# AVL tree

AVL tree
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In computer science, an AVL tree is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; at no time do they differ by more than one because rebalancing is done ensure this is the case. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.

AVL trees are often compared with red-black trees because they support the same set of operations and because red-black trees also take O(log n) time for the basic operations. Because AVL trees are more rigidly balanced, they are faster than red-black trees for lookup-intensive applications. Similar to red-black trees, AVL trees are height-balanced, but in general not weight-balanced nor μ-balanced; that is, sibling nodes can have hugely differing numbers of descendants.

Implement an AVL tree in the language of choice, and provide at least basic operations.

## Agda

This implementation uses the type system to enforce the height invariants, though not the BST invariants

 module Avl where -- The Peano naturalsdata Nat : Set where z : Nat s : Nat -> Nat -- An AVL tree's type is indexed by a natural.-- Avl N is the type of AVL trees of depth N. There arj 3 different-- node constructors:--   Left: The left subtree is one level deeper than the right--   Balanced: The subtrees have the same depth--   Right: The right Subtree is one level deeper than the left-- Since the AVL invariant is that the depths of a node's subtrees-- always differ by at most 1, this perfectly encodes the AVL depth invariant.data Avl : Nat -> Set where  Empty : Avl z  Left : {X : Nat} -> Nat -> Avl (s X) -> Avl X -> Avl (s (s X))  Balanced : {X : Nat} -> Nat -> Avl X -> Avl X -> Avl (s X)  Right : {X : Nat} -> Nat -> Avl X -> Avl (s X) -> Avl (s (s X)) -- A wrapper type that hides the AVL tree invariant. This is the interface-- exposed to the user.data Tree : Set where  avl : {N : Nat} -> Avl N -> Tree -- Comparison resultdata Ord : Set where  Less : Ord  Equal : Ord  Greater : Ord -- Comparison functioncmp : Nat -> Nat -> Ordcmp z (s X) = Lesscmp z z = Equalcmp (s X) z = Greatercmp (s X) (s Y) = cmp X Y -- Insertions can either leave the depth the same or-- increase it by one. Encode this in the type.data InsertResult : Nat -> Set where  Same : {X : Nat} -> Avl X -> InsertResult X  Bigger : {X : Nat} -> Avl (s X) -> InsertResult X -- If the left subtree is 2 levels deeper than the right, rotate to the right.-- balance-left X L R means X is the root, L is the left subtree and R is the right.balance-left : {N : Nat} -> Nat -> Avl (s (s N)) -> Avl N -> InsertResult (s (s N))balance-left X (Right Y A (Balanced Z B C)) D = Same (Balanced Z (Balanced X A B) (Balanced Y C D))balance-left X (Right Y A (Left Z B C)) D = Same (Balanced Z (Balanced X A B) (Right Y C D))balance-left X (Right Y A (Right Z B C)) D = Same (Balanced Z (Left X A B) (Balanced Y C D))balance-left X (Left Y (Balanced Z A B) C) D = Same (Balanced Z (Balanced X A B) (Balanced Y C D))balance-left X (Left Y (Left Z A B) C) D = Same (Balanced Z (Left X A B) (Balanced Y C D))balance-left X (Left Y (Right Z A B) C) D = Same (Balanced Z (Right X A B) (Balanced Y C D))balance-left X (Balanced Y (Balanced Z A B) C) D = Bigger (Right Z (Balanced X A B) (Left Y C D))balance-left X (Balanced Y (Left Z A B) C) D =  Bigger (Right Z (Left X A B) (Left Y C D))balance-left X (Balanced Y (Right Z A B) C) D = Bigger (Right Z (Right X A B) (Left Y C D)) -- Symmetric with balance-leftbalance-right : {N : Nat} -> Nat -> Avl N -> Avl (s (s N)) -> InsertResult (s (s N))balance-right X A (Left Y (Left Z B C) D) = Same (Balanced Z (Balanced X A B) (Right Y C D))balance-right X A (Left Y (Balanced Z B C) D) = Same(Balanced Z (Balanced  X A B) (Balanced Y C D))balance-right X A (Left Y (Right Z B C) D) = Same(Balanced Z (Left X A B) (Balanced Y C D))balance-right X A (Balanced Z B (Left Y C D)) = Bigger(Left Z (Right X A B) (Left Y C D))balance-right X A (Balanced Z B (Balanced Y C D)) = Bigger (Left Z (Right X A B) (Balanced Y C D))balance-right X A (Balanced Z B (Right Y C D)) = Bigger (Left Z (Right X A B) (Right Y C D))balance-right X A (Right Z B (Left Y C D)) = Same (Balanced Z (Balanced X A B) (Left Y C D))balance-right X A (Right Z B (Balanced Y C D)) = Same (Balanced Z (Balanced X A B) (Balanced Y C D))balance-right X A (Right Z B (Right Y C D)) =  Same (Balanced Z (Balanced X A B) (Right Y C D)) -- insert' T N does all the work of inserting the element N into the tree T.insert' : {N : Nat} -> Avl N -> Nat -> InsertResult Ninsert' Empty N = Bigger (Balanced N Empty Empty)insert' (Left Y L R) X with cmp X Yinsert' (Left Y L R) X | Less with insert' L Xinsert' (Left Y L R) X | Less | Same L' = Same (Left Y L' R)insert' (Left Y L R) X | Less | Bigger L' = balance-left Y L' Rinsert' (Left Y L R) X | Equal = Same (Left Y L R)insert' (Left Y L R) X | Greater with insert' R Xinsert' (Left Y L R) X | Greater | Same R' = Same (Left Y L R')insert' (Left Y L R) X | Greater | Bigger R' = Same (Balanced Y L R')insert' (Balanced Y L R) X with cmp X Yinsert' (Balanced Y L R) X | Less with insert' L Xinsert' (Balanced Y L R) X | Less | Same L'  = Same (Balanced Y L' R)insert' (Balanced Y L R) X | Less | Bigger L' = Bigger (Left Y L' R)insert' (Balanced Y L R) X | Equal = Same (Balanced Y L R)insert' (Balanced Y L R) X | Greater with insert' R Xinsert' (Balanced Y L R) X | Greater | Same R' = Same (Balanced Y L R')insert' (Balanced Y L R) X | Greater | Bigger R' = Bigger (Right Y L R')insert' (Right Y L R) X with cmp X Yinsert' (Right Y L R) X | Less with insert' L Xinsert' (Right Y L R) X | Less | Same L' = Same (Right Y L' R)insert' (Right Y L R) X | Less | Bigger L' = Same (Balanced Y L' R)insert' (Right Y L R) X | Equal = Same (Right Y L R)insert' (Right Y L R) X | Greater with insert' R Xinsert' (Right Y L R) X | Greater | Same R' = Same (Right Y L R')insert' (Right Y L R) X | Greater | Bigger R' = balance-right Y L R' -- Wrapper around insert' to use the depth-agnostic type Tree.insert : Tree -> Nat  -> Treeinsert (avl T) X with insert' T X... | Same T' = avl T'... | Bigger T' = avl T'

## beed

 eenioonneraashon staat{    heder,    balansd,    lepht_hi,    riit_hi} eenioonneraashon direcshon{  phronn_lepht,  phronn_riit} eenioonneraashon booleean{ troo, phals} clahs nohd<t>{    nohd<t> lepht;    nohd<t> riit;    nohd<t> pairent;    staat balans;    t daata;     nohd()    {       lepht = this;       riit = this;       balans = heder;    }     nohd(nohd<t> p, t daata_in)    {      pairent = p;      balans = balansd;      daata = daata_in;     }     is_heder(booleean anser)    {        balans.eecuuols(heder,anser);     }     eecuuols(nohd<t> other, booleean anser)    {        _nohd.nohd.eecuuols(other.nohd._nohd, anser);    }     nnoou_necst()    {        booleean is_heder = phals;        nohd.is_heder(is_heder);        iph (is_heder)        {            nohd = nohd.lepht;            reeturn;        }         booleean riit_is_null = phals;        nohd.riit.is_nul(riit_is_null);        iph (!riit_is_null)        {           nohd = nohd.riit;            booleean lepht_is_nul = phals;           nohd.lepht.is_nul(lepht_is_nul);           uuhiil (!lepht_is_null)           {               nohd = nohd.lepht;               nohd.lepht.is_nul(lepht_is_nul);           }        }        els        {            nohd<t> uui = nohd.pairent;            booleean uui_is_heder = phals;            uui.is_heder(uui_is_heder);            iph (uui_is_heder)            {                nohd = uui;                reeturn;            }            booleean nohd_is_uui_riit = phals;            nohd.eecuuols(uui.riit,nohd_is_uui_riit);            uuhiil (nohd_is_uui_riit)            {                nohd = uui;                uui = uui.pairent;                nohd.eecuuols(uui.riit,nohd_is_uui_riit);            }            nohd = uui;        }     }     rohtaat_lepht(nohd<t> root)    {        nohd<t> pairent = root.pairent;        nohd<t> e = root.riit;        root.pairent = e;        e.pairent = pairent;        booleean e_lepht_is_nul = phals;        nohd<t> e_lepht = e.lepht;        e_lepht.is_nul(e_lepht_is_nul);        iph (!e_lepht_is_nul) {e.lepht.pairent = root;}        root.riit = e.lepht;        e.lepht = root;        root = e;    }     rohtaat_riit(nohd<t> root)    {        nohd<t> pairent = root.pairent;        nohd<t> e = root.lepht;        root.pairent = e;        e.pairent = pairent;        booleean e_riit_is_nul = phals;        nohd<t> e_riit =  e.riit;        e_riit.is_nul(e_lepht_is_nul);        iph (!e_lepht_is_nul) {e.riit.pairent = root;}        root.lepht = e.riit;        e.riit = root;        root = e;    }     balans_lepht(nohd<t> root)    {        nohd<t> lepht = root.lepht;        select (lepht.balans)        {            caas lepht_hi            {                root.balans = balansd;                lepht.balans = balansd;                rohtaat_riit(root);            }             caas riit_hi            {                nohd<t> subriit = lepht.riit;                 select(subriit.balans)                {                    caas balansd                    {                        root.balans = balansd;                        lepht.balans = balansd;                    }                     caas riit_hi                    {                        root.balans = balansd;                        lepht.balans = lepht_hi;                    }                     caas lepht_hi                    {                        root.balans = riit_hi;                        lepht.balans = balansd;                    }                }                subriit.balans = balansd;                rohtaat_lepht(lepht);                root.lepht = lepht;                rohtaat_riit(root);            }             caas balansd            {                root.balans = lepht_hi;                lepht.balans = riit_hi;                rohtaat_riit(root);            }        }       }     balans_riit(nohd<t> root)    {        nohd<t> riit = root.riit;        select (riit.balans)        {            caas riit_hi            {                 root.balans = balansd;                riit.balans = balansd;                rohtaat_lepht(root);            }            caas lepht_hi           {               nohd<t> sublepht = riit.lepht;                select(sublepht.balans)               {                   caas balansd                   {                       root.balans = balansd;                       riit.balans = balansd;                   }                    caas lepht_hi                   {                       root.balans = balansd;                       riit.balans = riit_hi;                   }                    caas riit_hi                   {                       root.balans = lepht_hi;                       riit.balans = balansd;                   }               }               sublepht.balans = balansd;               rohtaat_riit(riit);               root.riit = riit;               rohtaat_lepht(root);           }            caas balansd           {               root.balans = riit_hi;               riit.balans = lepht_hi;               rohtaat_lepht(root);           }        }       }     balans_tree(nohd<t> root, direcshon phronn)    {        booleean torler = troo;         uuhiil (torler)        {            nohd<t> pairent = root.pairent;             direcshon necst_phronn = phronn_lepht;             booleean is_pairent_lepht = phals;            pairent.lepht.eecuuols(root,is_pairent_lepht);            iph (!is_pairent_lepht)            {                necst_phronn = phronn_riit;            }             booleean is_phronn_lepht = phals;            phronn.eecuuols(phronn_lepht, is_phronn_lepht);            iph (is_phronn_lepht)            {                select (root.balans)                {                    caas lepht_hi                    {                        booleean pheder = phals;                        pairent.is_heder(pheder);                        iph (pheder)                        {                            balans_lepht(pairent.pairent);                        }                        els                        {                            booleean lepht_is_root = phals;                            pairent.lepht.eecuuols(root,lepht_is_root);                            iph (lepht_is_root)                            {                                balans_lepht(pairent.lepht);                            }                            els                            {                                balans_lepht = pairent.riit;                                torler = phals;                            }                        }                    }                     caas balansd                    {                        root.balans = lepht_hi;                        torler = troo;                    }                     caas riit_hi                    {                        root.balans = balansd;                        torler = phals;                    }                }            }            els            {               select (root.balans)               {                   caas lepht_hi                   {                       root.balans = balansd;                       torler = phals;                   }                        caas balansd                   {                       root.balans = riit_hi;                       torler = troo;                   }                    caas riit_hi                   {                        booleean heder = phals;                        pairent.is_heder(heder);                        iph (heder)                        {                            balans_riit(pairent.pairent);                        }                        els                        {                            booleean lpairent = phals;                            pairent.lepht.eecuuols(root,lpairent);                            iph (lpairent)                            {                                balans_riit(pairent.lepht);                            }                            els                            {                                balans_riit(pairent.riit);                                torler = phals;                            }                        }                     }                 }             }              iph (torler)             {                 booleean heder_up = phals;                 pairent.is_heder(heder_up);                 iph (heder_up)                 {                   torler = phals;                  }                  els                  {                      root = pairent;                      phronn = necst_phronn;                  }             }         }    }     balans_tree_reennoou(nohd<t> root, direcshon phronn)    {         booleean shorter = troo;          uuhiil (shorter)         {            nohd<t> pairent = root.pairent;             direcshon necst_phronn = phronn_lepht;             booleean is_pairent_lepht = phals;            pairent.lepht.eecuuols(root,is_pairent_lepht);            iph (!is_pairent_lepht)            {                necst_phronn = phronn_riit;            }             booleean is_phronn_lepht = phals;            phronn.eecuuols(phronn_lepht, is_phronn_lepht);            iph (is_phronn_lepht)            {                select (root.balans)                {                    caas lepht_hi                    {                        root.balans = staat.balansd;                        shorter = troo;                    }                      caas balansd                    {                        root.balans = staat.riit_hi;                        shorter = phals;                    }                     caas riit_hi                    {                        booleean riit_is_balansd = phals;                        root.riit.balans.eecuuols(balansd,riit_is_balansd);                         iph (riit_is_balansd)                        {                            shorter = phals;                        }                        els                        {                            shorter = troo;                        }                         booleean heder = phals;                        pairent.is_heder(heder);                        iph (heder)                        {                            balans_riit(pairent.pairent);                        }                        els                        {                            booleean lpairent = phals;                            pairent.lepht.eecuuols(root,lpairent);                            iph (lpairent)                            {                                balans_riit(pairent.lepht);                            }                            els                            {                                balans_riit(pairent.riit);                            }                       }                       }                }             }             els             {                select (root.balans)                {                    caas riit_hi                    {                        root.balans = balansd;                        shorter = troo;                    }                     caas balansd                   {                      root.balans = lepht_hi;                      shorter = phals;                   }                     caas lepht_hi                   {                        booleean lepht_is_balansd = phals;                        root.lepht.balans.eecuuols(balansd,lepht_is_balansd);                        iph (lepht_is_balansd)                        {                            shorter = phals;                        }                        els                        {                            shorter = troo;                        }                         booleean is_heder = phals;                        pairent.is_heder(is_heder);                        iph (is_heder)                        {                            balans_lepht(pairent.pairent);                        }                        els                        {                            pairent.lepht.eecuuols(root,lpairent);                            iph (lpairent)                            {                                balans_lepht(pairent.lepht);                            }                            els                            {                                balans_lepht(pairent.riit);                            }                       }                   }                }            }             iph (shorter)            {               booleean heder_up = phals;               pairent.is_heder(heder_up);               iph (heder_up)               {                  shorter = phals;               }               els               {                  root = pairent;                  phronn = necst_phronn;               }           }        }    }}   clahs entree_orlredee_ecsists_ecssepshon{   string naann;     entree_orlredee_ecsists_ecssepshon(string naann_in)    {         naann = naann_in;    }     print()    {         string ouut(naan);         string to_ad( orlredee ecsists);         ouut.concat(to_ad);         ouut.print();    }}  } clahs set_entree<t> {    set_entree(nohd<t> n)    {        _nohd = n;    }     ualioo(t _daata)    {      _daata = _nohd.daata;     }     is_heder(booleean b) { _nohd.nohd.is_heder(b); }      nohd<t> _nohd;} clahs set<t>{    nohd<t> heder;     set()    {       heder();    }     ad(t daata)    {       string out(in ad);       out.println();         booleean root_is_nul = phals;        heder.pairent.is_nul(root_is_nul);        iph (root_is_nul)        {            nohd<t> n(heder,daata);            heder.pairent = n;            heder.lepht = heder.pairent;            heder.riit = heder.pairent;        }        els        {            nohd<t> root = heder.pairent;             reepeet            {                booleean is_les = phals;                string outb(connpairing entrees);                outb.println();                daata.les(root.daata, is_les);                iph (is_les)                {                    booleean root_lepht_is_nul = phals;                    root.nohd.lepht.is_nul(root_lepht_is_nul);                    iph (!root_lepht_is_nul)                    {                        root.nohd = root.lepht;                    }                    els                    {                        nohd<t> nioo_nohd(root,daata);                        root.lepht = nioo_nohd;                        booleean is_phurst = phals;                        heder.lepht.eecuuols(root,is_phurst);                        iph (is_phurst)                        {                            heder.lepht = nioo_nohd;                        }                        direcshon dir(phronn_lepht);                        balans_tree(root.nohd, dir);                        reeturn;                    }               }               els               {                    booleean is_graater = phals;                    root.daata.les(daata, is_graater);                    iph (is_graater)                    {                          booleean root_riit_is_nul = phals;                        root.nohd.riit.is_nul(root_riit_is_nul);                        iph (!root_riit_is_nul)                        {                            root = root.riit;                        }                        els                        {                            nohd<t> nioo(root,daata);                            root.riit = nioo;                            booleean is_lahst = phals;                            heder.riit.eecuuols(root,is_lahst);                            iph (is_lahst)                            {                                heder.riit = nioo_nohd;                            }                            direcshon dirb(phronn_riit);                            balans_tree(root.nohd, dirb);                            reeturn;                        }                    }                    els // iiten orlredee ecsists                    {                        string s(entree orlredee ecsists);                        entree_orlredee_ecsists_ecssepshon p(s);                        throuu p;                    }               }           }       }   }}

See AVL tree/C

## C#

See AVL_tree/C_sharp.

## C++

Translation of: D
 #include <algorithm>#include <iostream> /* AVL node */template <class T>class AVLnode {public:    T key;    int balance;    AVLnode *left, *right, *parent;     AVLnode(T k, AVLnode *p) : key(k), balance(0), parent(p),                        left(NULL), right(NULL) {}     ~AVLnode() {        delete left;        delete right;    }}; /* AVL tree */template <class T>class AVLtree {public:    AVLtree(void);    ~AVLtree(void);    bool insert(T key);    void deleteKey(const T key);    void printBalance(); private:    AVLnode<T> *root;     AVLnode<T>* rotateLeft          ( AVLnode<T> *a );    AVLnode<T>* rotateRight         ( AVLnode<T> *a );    AVLnode<T>* rotateLeftThenRight ( AVLnode<T> *n );    AVLnode<T>* rotateRightThenLeft ( AVLnode<T> *n );    void rebalance                  ( AVLnode<T> *n );    int height                      ( AVLnode<T> *n );    void setBalance                 ( AVLnode<T> *n );    void printBalance               ( AVLnode<T> *n );    void clearNode                  ( AVLnode<T> *n );}; /* AVL class definition */template <class T>void AVLtree<T>::rebalance(AVLnode<T> *n) {    setBalance(n);     if (n->balance == -2) {        if (height(n->left->left) >= height(n->left->right))            n = rotateRight(n);        else            n = rotateLeftThenRight(n);    }    else if (n->balance == 2) {        if (height(n->right->right) >= height(n->right->left))            n = rotateLeft(n);        else            n = rotateRightThenLeft(n);    }     if (n->parent != NULL) {        rebalance(n->parent);    }    else {        root = n;    }} template <class T>AVLnode<T>* AVLtree<T>::rotateLeft(AVLnode<T> *a) {    AVLnode<T> *b = a->right;    b->parent = a->parent;    a->right = b->left;     if (a->right != NULL)        a->right->parent = a;     b->left = a;    a->parent = b;     if (b->parent != NULL) {        if (b->parent->right == a) {            b->parent->right = b;        }        else {            b->parent->left = b;        }    }     setBalance(a);    setBalance(b);    return b;} template <class T>AVLnode<T>* AVLtree<T>::rotateRight(AVLnode<T> *a) {    AVLnode<T> *b = a->left;    b->parent = a->parent;    a->left = b->right;     if (a->left != NULL)        a->left->parent = a;     b->right = a;    a->parent = b;     if (b->parent != NULL) {        if (b->parent->right == a) {            b->parent->right = b;        }        else {            b->parent->left = b;        }    }     setBalance(a);    setBalance(b);    return b;} template <class T>AVLnode<T>* AVLtree<T>::rotateLeftThenRight(AVLnode<T> *n) {    n->left = rotateLeft(n->left);    return rotateRight(n);} template <class T>AVLnode<T>* AVLtree<T>::rotateRightThenLeft(AVLnode<T> *n) {    n->right = rotateRight(n->right);    return rotateLeft(n);} template <class T>int AVLtree<T>::height(AVLnode<T> *n) {    if (n == NULL)        return -1;    return 1 + std::max(height(n->left), height(n->right));} template <class T>void AVLtree<T>::setBalance(AVLnode<T> *n) {    n->balance = height(n->right) - height(n->left);} template <class T>void AVLtree<T>::printBalance(AVLnode<T> *n) {    if (n != NULL) {        printBalance(n->left);        std::cout << n->balance << " ";        printBalance(n->right);    }} template <class T>AVLtree<T>::AVLtree(void) : root(NULL) {} template <class T>AVLtree<T>::~AVLtree(void) {    delete root;} template <class T>bool AVLtree<T>::insert(T key) {    if (root == NULL) {        root = new AVLnode<T>(key, NULL);    }    else {        AVLnode<T>            *n = root,            *parent;         while (true) {            if (n->key == key)                return false;             parent = n;             bool goLeft = n->key > key;            n = goLeft ? n->left : n->right;             if (n == NULL) {                if (goLeft) {                    parent->left = new AVLnode<T>(key, parent);                }                else {                    parent->right = new AVLnode<T>(key, parent);                }                 rebalance(parent);                break;            }        }    }     return true;} template <class T>void AVLtree<T>::deleteKey(const T delKey) {    if (root == NULL)        return;     AVLnode<T>        *n       = root,        *parent  = root,        *delNode = NULL,        *child   = root;     while (child != NULL) {        parent = n;        n = child;        child = delKey >= n->key ? n->right : n->left;        if (delKey == n->key)            delNode = n;    }     if (delNode != NULL) {        delNode->key = n->key;         child = n->left != NULL ? n->left : n->right;         if (root->key == delKey) {            root = child;        }        else {            if (parent->left == n) {                parent->left = child;            }            else {                parent->right = child;            }             rebalance(parent);        }    }} template <class T>void AVLtree<T>::printBalance() {    printBalance(root);    std::cout << std::endl;} int main(void){    AVLtree<int> t;     std::cout << "Inserting integer values 1 to 10" << std::endl;    for (int i = 1; i <= 10; ++i)        t.insert(i);     std::cout << "Printing balance: ";    t.printBalance();}
Output:
Inserting integer values 1 to 10
Printing balance: 0 0 0 1 0 0 0 0 1 0


See AVL_tree/C++

### Managed C++

See AVL_tree/Managed_C++ </lang>

## Common Lisp

Provided is an imperative implementation of an AVL tree with a similar interface and documentation to HASH-TABLE.

(defpackage :avl-tree  (:use :cl)  (:export   :avl-tree   :make-avl-tree   :avl-tree-count   :avl-tree-p   :avl-tree-key<=   :gettree   :remtree   :clrtree   :dfs-maptree   :bfs-maptree)) (in-package :avl-tree) (defstruct %tree  key  value  (height 0 :type fixnum)  left  right) (defstruct (avl-tree (:constructor %make-avl-tree))  key<=  tree  (count 0 :type fixnum)) (defun make-avl-tree (key<=)  "Create a new AVL tree using the given comparison function KEY<=for emplacing keys into the tree."  (%make-avl-tree :key<= key<=)) (declaim (inline          recalc-height          height balance          swap-kv          right-right-rotate          right-left-rotate          left-right-rotate          left-left-rotate          rotate)) (defun recalc-height (tree)  "Calculate the new height of the tree from the heights of the children."  (when tree    (setf (%tree-height tree)          (1+ (the fixnum (max (height (%tree-right tree))                               (height (%tree-left tree)))))))) (declaim (ftype (function (t) fixnum) height balance))(defun height (tree)  (if tree (%tree-height tree) 0)) (defun balance (tree)  (if tree      (- (height (%tree-right tree))         (height (%tree-left tree)))      0)) (defmacro swap (place-a place-b)  "Swap the values of two places."  (let ((tmp (gensym)))    (let ((,tmp ,place-a))       (setf ,place-a ,place-b)       (setf ,place-b ,tmp)))) (defun swap-kv (tree-a tree-b)  "Swap the keys and values of two trees."  (swap (%tree-value tree-a) (%tree-value tree-b))  (swap (%tree-key tree-a) (%tree-key tree-b))) ;; We should really use gensyms for the variables in here.(defmacro slash-rotate (tree right left)  "Rotate nodes in a slash / imbalance."  (let* ((a ,tree)          (b (,right a))          (c (,right b))          (a-left (,left a))          (b-left (,left b)))     (setf (,right a) c)     (setf (,left a) b)     (setf (,left b) a-left)     (setf (,right b) b-left)     (swap-kv a b)     (recalc-height b)     (recalc-height a))) (defmacro angle-rotate (tree right left)  "Rotate nodes in an angle bracket < imbalance."  (let* ((a ,tree)          (b (,right a))          (c (,left b))          (a-left (,left a))          (c-left (,left c))          (c-right (,right c)))     (setf (,left a) c)     (setf (,left c) a-left)     (setf (,right c) c-left)     (setf (,left b) c-right)     (swap-kv a c)     (recalc-height c)     (recalc-height b)     (recalc-height a))) (defun right-right-rotate (tree)  (slash-rotate tree %tree-right %tree-left)) (defun left-left-rotate (tree)  (slash-rotate tree %tree-left %tree-right)) (defun right-left-rotate (tree)  (angle-rotate tree %tree-right %tree-left)) (defun left-right-rotate (tree)  (angle-rotate tree %tree-left %tree-right)) (defun rotate (tree)  (declare (type %tree tree))  "Perform a rotation on the given TREE if it is imbalanced."  (recalc-height tree)  (with-slots (left right) tree    (let ((balance (balance tree)))      (cond ((< 1 balance) ;; Right imbalanced tree             (if (<= 0 (balance right))                 (right-right-rotate tree)                 (right-left-rotate tree)))            ((> -1 balance) ;; Left imbalanced tree             (if (<= 0 (balance left))                 (left-right-rotate tree)                 (left-left-rotate tree))))))) (defun gettree (key avl-tree &optional default)  "Finds an entry in AVL-TREE whos key is KEY and returns theassociated value and T as multiple values, or returns DEFAULT and NILif there was no such entry. Entries can be added using SETF."  (with-slots (key<= tree) avl-tree    (labels        ((rec (tree)           (if tree               (with-slots ((t-key key) left right value) tree                 (if (funcall key<= t-key key)                     (if (funcall key<= key t-key)                         (values value t)                         (rec right))                     (rec left)))               (values default nil))))      (rec tree)))) (defun puttree (value key avl-tree)  ;;(declare (optimize speed))  (declare (type avl-tree avl-tree))  "Emplace the the VALUE with the given KEY into the AVL-TREE, oroverwrite the value if the given key already exists."  (let ((node (make-%tree :key key :value value)))    (with-slots (key<= tree count) avl-tree      (cond (tree             (labels                 ((rec (tree)                    (with-slots ((t-key key) left right) tree                      (if (funcall key<= t-key key)                          (if (funcall key<= key t-key)                              (setf (%tree-value tree) value)                              (cond (right (rec right))                                    (t (setf right node)                                       (incf count))))                          (cond (left (rec left))                                (t (setf left node)                                   (incf count))))                      (rotate tree))))               (rec tree)))            (t (setf tree node)               (incf count))))    value)) (defun (setf gettree) (value key avl-tree &optional default)  (declare (ignore default))  (puttree value key avl-tree)) (defun remtree (key avl-tree)  (declare (type avl-tree avl-tree))  "Remove the entry in AVL-TREE associated with KEY. Return T ifthere was such an entry, or NIL if not."  (with-slots (key<= tree count) avl-tree    (labels        ((find-left (tree)           (with-slots ((t-key key) left right) tree             (if left                 (find-left left)                 tree)))         (rec (tree &optional parent type)           (when tree             (prog1                 (with-slots ((t-key key) left right) tree                   (if (funcall key<= t-key key)                       (cond                         ((funcall key<= key t-key)                          (cond                            ((and left right)                             (let ((sub-left (find-left right)))                               (swap-kv sub-left tree)                               (rec right tree :right)))                            (t                             (let ((sub (or left right)))                               (case type                                 (:right (setf (%tree-right parent) sub))                                 (:left (setf (%tree-left parent) sub))                                 (nil (setf (avl-tree-tree avl-tree) sub))))                             (decf count)))                          t)                         (t (rec right tree :right)))                       (rec left tree :left)))               (when parent (rotate parent))))))      (rec tree)))) (defun clrtree (avl-tree)  "This removes all the entries from AVL-TREE and returns the tree itself."  (setf (avl-tree-tree avl-tree) nil)  (setf (avl-tree-count avl-tree) 0)  avl-tree) (defun dfs-maptree (function avl-tree)  "For each entry in AVL-TREE call the two-argument FUNCTION onthe key and value of each entry in depth-first order from left to right.Consequences are undefined if AVL-TREE is modified during this call."  (with-slots (key<= tree) avl-tree    (labels        ((rec (tree)           (when tree             (with-slots ((t-key key) left right key value) tree               (rec left)               (funcall function key value)               (rec right)))))      (rec tree)))) (defun bfs-maptree (function avl-tree)  "For each entry in AVL-TREE call the two-argument FUNCTION onthe key and value of each entry in breadth-first order from left to right.Consequences are undefined if AVL-TREE is modified during this call."  (with-slots (key<= tree) avl-tree    (let* ((queue (cons nil nil))           (end queue))      (labels ((pushend (value)                 (when value                   (setf (cdr end) (cons value nil))                   (setf end (cdr end))))               (empty-p () (eq nil (cdr queue)))               (popfront ()                 (prog1 (pop (cdr queue))                   (when (empty-p) (setf end queue)))))        (when tree          (pushend tree)          (loop until (empty-p)             do (let ((current (popfront)))                  (with-slots (key value left right) current                    (funcall function key value)                    (pushend left)                    (pushend right))))))))) (defun test ()  (let ((tree (make-avl-tree #'<=))        (printer (lambda (k v) (print (list k v)))))    (loop for key in '(0 8 6 4 2 3 7 9 1 5 5)       for value in '(a b c d e f g h i j k)       do (setf (gettree key tree) value))    (loop for key in '(0 1 2 3 4 10)       do (print (multiple-value-list (gettree key tree))))    (terpri)    (print tree)    (terpri)    (dfs-maptree printer tree)    (terpri)    (bfs-maptree printer tree)    (terpri)    (loop for key in '(0 1 2 3 10 7)       do (print (remtree key tree)))    (terpri)    (print tree)    (terpri)    (clrtree tree)    (print tree))  (values)) (defun profile-test ()  (let ((tree (make-avl-tree #'<=))        (randoms (loop repeat 1000000 collect (random 100.0))))    (loop for key in randoms do (setf (gettree key tree) key))))

## D

Translation of: Java
import std.stdio, std.algorithm; class AVLtree {    private Node* root;     private static struct Node {        private int key, balance;        private Node* left, right, parent;         this(in int k, Node* p) pure nothrow @safe @nogc {            key = k;            parent = p;        }    }     public bool insert(in int key) pure nothrow @safe {        if (root is null)            root = new Node(key, null);        else {            Node* n = root;            Node* parent;            while (true) {                if (n.key == key)                    return false;                 parent = n;                 bool goLeft = n.key > key;                n = goLeft ? n.left : n.right;                 if (n is null) {                    if (goLeft) {                        parent.left = new Node(key, parent);                    } else {                        parent.right = new Node(key, parent);                    }                    rebalance(parent);                    break;                }            }        }        return true;    }     public void deleteKey(in int delKey) pure nothrow @safe @nogc {        if (root is null)            return;        Node* n = root;        Node* parent = root;        Node* delNode = null;        Node* child = root;         while (child !is null) {            parent = n;            n = child;            child = delKey >= n.key ? n.right : n.left;            if (delKey == n.key)                delNode = n;        }         if (delNode !is null) {            delNode.key = n.key;             child = n.left !is null ? n.left : n.right;             if (root.key == delKey) {                root = child;            } else {                if (parent.left is n) {                    parent.left = child;                } else {                    parent.right = child;                }                rebalance(parent);            }        }    }     private void rebalance(Node* n) pure nothrow @safe @nogc {        setBalance(n);         if (n.balance == -2) {            if (height(n.left.left) >= height(n.left.right))                n = rotateRight(n);            else                n = rotateLeftThenRight(n);         } else if (n.balance == 2) {            if (height(n.right.right) >= height(n.right.left))                n = rotateLeft(n);            else                n = rotateRightThenLeft(n);        }         if (n.parent !is null) {            rebalance(n.parent);        } else {            root = n;        }    }     private Node* rotateLeft(Node* a) pure nothrow @safe @nogc {        Node* b = a.right;        b.parent = a.parent;         a.right = b.left;         if (a.right !is null)            a.right.parent = a;         b.left = a;        a.parent = b;         if (b.parent !is null) {            if (b.parent.right is a) {                b.parent.right = b;            } else {                b.parent.left = b;            }        }         setBalance(a, b);         return b;    }     private Node* rotateRight(Node* a) pure nothrow @safe @nogc {        Node* b = a.left;        b.parent = a.parent;         a.left = b.right;         if (a.left !is null)            a.left.parent = a;         b.right = a;        a.parent = b;         if (b.parent !is null) {            if (b.parent.right is a) {                b.parent.right = b;            } else {                b.parent.left = b;            }        }         setBalance(a, b);         return b;    }     private Node* rotateLeftThenRight(Node* n) pure nothrow @safe @nogc {        n.left = rotateLeft(n.left);        return rotateRight(n);    }     private Node* rotateRightThenLeft(Node* n) pure nothrow @safe @nogc {        n.right = rotateRight(n.right);        return rotateLeft(n);    }     private int height(in Node* n) const pure nothrow @safe @nogc {        if (n is null)            return -1;        return 1 + max(height(n.left), height(n.right));    }     private void setBalance(Node*[] nodes...) pure nothrow @safe @nogc {        foreach (n; nodes)            n.balance = height(n.right) - height(n.left);    }     public void printBalance() const @safe {        printBalance(root);    }     private void printBalance(in Node* n) const @safe {        if (n !is null) {            printBalance(n.left);            write(n.balance, ' ');            printBalance(n.right);        }    }} void main() @safe {    auto tree = new AVLtree();     writeln("Inserting values 1 to 10");    foreach (immutable i; 1 .. 11)        tree.insert(i);     write("Printing balance: ");    tree.printBalance;}
Output:
Inserting values 1 to 10
Printing balance: 0 0 0 1 0 0 0 0 1 0 

## Go

A package:

package avl // AVL tree adapted from Julienne Walker's presentation at// http://eternallyconfuzzled.com/tuts/datastructures/jsw_tut_avl.aspx.// This port uses similar indentifier names. // The Key interface must be supported by data stored in the AVL tree.type Key interface {    Less(Key) bool    Eq(Key) bool} // Node is a node in an AVL tree.type Node struct {    Data    Key      // anything comparable with Less and Eq.    Balance int      // balance factor    Link    [2]*Node // children, indexed by "direction", 0 or 1.} // A little readability function for returning the opposite of a direction,// where a direction is 0 or 1.  Go inlines this.// Where JW writes !dir, this code has opp(dir).func opp(dir int) int {    return 1 - dir} // single rotationfunc single(root *Node, dir int) *Node {    save := root.Link[opp(dir)]    root.Link[opp(dir)] = save.Link[dir]    save.Link[dir] = root    return save} // double rotationfunc double(root *Node, dir int) *Node {    save := root.Link[opp(dir)].Link[dir]     root.Link[opp(dir)].Link[dir] = save.Link[opp(dir)]    save.Link[opp(dir)] = root.Link[opp(dir)]    root.Link[opp(dir)] = save     save = root.Link[opp(dir)]    root.Link[opp(dir)] = save.Link[dir]    save.Link[dir] = root    return save} // adjust valance factors after double rotationfunc adjustBalance(root *Node, dir, bal int) {    n := root.Link[dir]    nn := n.Link[opp(dir)]    switch nn.Balance {    case 0:        root.Balance = 0        n.Balance = 0    case bal:        root.Balance = -bal        n.Balance = 0    default:        root.Balance = 0        n.Balance = bal    }    nn.Balance = 0} func insertBalance(root *Node, dir int) *Node {    n := root.Link[dir]    bal := 2*dir - 1    if n.Balance == bal {        root.Balance = 0        n.Balance = 0        return single(root, opp(dir))    }    adjustBalance(root, dir, bal)    return double(root, opp(dir))} func insertR(root *Node, data Key) (*Node, bool) {    if root == nil {        return &Node{Data: data}, false    }    dir := 0    if root.Data.Less(data) {        dir = 1    }    var done bool    root.Link[dir], done = insertR(root.Link[dir], data)    if done {        return root, true    }    root.Balance += 2*dir - 1    switch root.Balance {    case 0:        return root, true    case 1, -1:        return root, false    }    return insertBalance(root, dir), true} // Insert a node into the AVL tree.// Data is inserted even if other data with the same key already exists.func Insert(tree **Node, data Key) {    *tree, _ = insertR(*tree, data)} func removeBalance(root *Node, dir int) (*Node, bool) {    n := root.Link[opp(dir)]    bal := 2*dir - 1    switch n.Balance {    case -bal:        root.Balance = 0        n.Balance = 0        return single(root, dir), false    case bal:        adjustBalance(root, opp(dir), -bal)        return double(root, dir), false    }    root.Balance = -bal    n.Balance = bal    return single(root, dir), true} func removeR(root *Node, data Key) (*Node, bool) {    if root == nil {        return nil, false    }    if root.Data.Eq(data) {        switch {        case root.Link[0] == nil:            return root.Link[1], false        case root.Link[1] == nil:            return root.Link[0], false        }        heir := root.Link[0]        for heir.Link[1] != nil {            heir = heir.Link[1]        }        root.Data = heir.Data        data = heir.Data    }    dir := 0    if root.Data.Less(data) {        dir = 1    }    var done bool    root.Link[dir], done = removeR(root.Link[dir], data)    if done {        return root, true    }    root.Balance += 1 - 2*dir    switch root.Balance {    case 1, -1:        return root, true    case 0:        return root, false    }    return removeBalance(root, dir)} // Remove a single item from an AVL tree.// If key does not exist, function has no effect.func Remove(tree **Node, data Key) {    *tree, _ = removeR(*tree, data)}

A demonstration program:

package main import (    "encoding/json"    "fmt"    "log"     "avl") type intKey int // satisfy avl.Keyfunc (k intKey) Less(k2 avl.Key) bool { return k < k2.(intKey) }func (k intKey) Eq(k2 avl.Key) bool   { return k == k2.(intKey) } // use json for cheap tree visualizationfunc dump(tree *avl.Node) {    b, err := json.MarshalIndent(tree, "", "   ")    if err != nil {        log.Fatal(err)    }    fmt.Println(string(b))} func main() {    var tree *avl.Node    fmt.Println("Empty tree:")    dump(tree)     fmt.Println("\nInsert test:")    avl.Insert(&tree, intKey(3))    avl.Insert(&tree, intKey(1))    avl.Insert(&tree, intKey(4))    avl.Insert(&tree, intKey(1))    avl.Insert(&tree, intKey(5))    dump(tree)     fmt.Println("\nRemove test:")    avl.Remove(&tree, intKey(3))    avl.Remove(&tree, intKey(1))    dump(tree)}
Output:
Empty tree:
null

Insert test:
{
"Data": 3,
"Balance": 0,
{
"Data": 1,
"Balance": -1,
{
"Data": 1,
"Balance": 0,
null,
null
]
},
null
]
},
{
"Data": 4,
"Balance": 1,
null,
{
"Data": 5,
"Balance": 0,
null,
null
]
}
]
}
]
}

Remove test:
{
"Data": 4,
"Balance": 0,
{
"Data": 1,
"Balance": 0,
null,
null
]
},
{
"Data": 5,
"Balance": 0,
null,
null
]
}
]
}


Based on solution of homework #4 from course http://www.seas.upenn.edu/~cis194/spring13/lectures.html.

import Data.Monoid data Tree a  = Leaf  | Node Int         (Tree a)         a         (Tree a)  deriving (Show, Eq) foldTree  :: Ord a  => [a] -> Tree afoldTree = foldr insert Leaf height Leaf = -1height (Node h _ _ _) = h depth a b = 1 + (height a max height b) insert  :: Ord a  => a -> Tree a -> Tree ainsert v Leaf = Node 1 Leaf v Leafinsert v t@(Node n left v_ right)  | v_ < v = rotate $Node n left v_ (insert v right) | v_ > v = rotate$ Node n (insert v left) v_ right  | otherwise = t max_  :: Ord a  => Tree a -> Maybe amax_ Leaf = Nothingmax_ (Node _ _ v right) =  case right of    Leaf -> Just v    _ -> max_ right delete  :: Ord a  => a -> Tree a -> Tree adelete _ Leaf = Leafdelete x (Node h left v right)  | x == v =    maybe left (\m -> rotate $Node h left m (delete m right)) (max_ right) | x > v = rotate$ Node h left v (delete x right)  | x < v = rotate $Node h (delete x left) v right rotate :: Tree a -> Tree arotate Leaf = Leaf-- left left caserotate (Node h (Node lh ll lv lr) v r) | lh - height r > 1 && height ll - height lr > 0 = Node lh ll lv (Node (depth r lr) lr v r)-- right right caserotate (Node h l v (Node rh rl rv rr)) | rh - height l > 1 && height rr - height rl > 0 = Node rh (Node (depth l rl) l v rl) rv rr-- left right caserotate (Node h (Node lh ll lv (Node rh rl rv rr)) v r) | lh - height r > 1 = Node h (Node (rh + 1) (Node (lh - 1) ll lv rl) rv rr) v r-- right left caserotate (Node h l v (Node rh (Node lh ll lv lr) rv rr)) | rh - height l > 1 = Node h l v (Node (lh + 1) ll lv (Node (rh - 1) lr rv rr))-- re-weightingrotate (Node h l v r) = let (l_, r_) = (rotate l, rotate r) in Node (depth l_ r_) l_ v r_ draw :: Show a => Tree a -> Stringdraw t = '\n' : draw_ t 0 <> "\n" where draw_ Leaf _ = [] draw_ (Node h l v r) d = draw_ r (d + 1) <> node <> draw_ l (d + 1) where node = padding d <> show (v, h) <> "\n" padding n = replicate (n * 4) ' ' main :: IO ()main = putStr$ draw $foldTree [1 .. 15] Output:  (15,0) (14,1) (13,0) (12,2) (11,0) (10,1) (9,0) (8,3) (7,0) (6,1) (5,0) (4,2) (3,0) (2,1) (1,0)  ## Java This code has been cobbled together from various online examples. It's not easy to find a clear and complete explanation of AVL trees. Textbooks tend to concentrate on red-black trees because of their better efficiency. (AVL trees need to make 2 passes through the tree when inserting and deleting: one down to find the node to operate upon and one up to rebalance the tree.) public class AVLtree { private Node root; private static class Node { private int key; private int balance; private int height; private Node left; private Node right; private Node parent; Node(int key, Node parent) { this.key = key; this.parent = parent; } } public boolean insert(int key) { if (root == null) { root = new Node(key, null); return true; } Node n = root; while (true) { if (n.key == key) return false; Node parent = n; boolean goLeft = n.key > key; n = goLeft ? n.left : n.right; if (n == null) { if (goLeft) { parent.left = new Node(key, parent); } else { parent.right = new Node(key, parent); } rebalance(parent); break; } } return true; } private void delete(Node node) { if (node.left == null && node.right == null) { if (node.parent == null) { root = null; } else { Node parent = node.parent; if (parent.left == node) { parent.left = null; } else { parent.right = null; } rebalance(parent); } return; } if (node.left != null) { Node child = node.left; while (child.right != null) child = child.right; node.key = child.key; delete(child); } else { Node child = node.right; while (child.left != null) child = child.left; node.key = child.key; delete(child); } } public void delete(int delKey) { if (root == null) return; Node child = root; while (child != null) { Node node = child; child = delKey >= node.key ? node.right : node.left; if (delKey == node.key) { delete(node); return; } } } private void rebalance(Node n) { setBalance(n); if (n.balance == -2) { if (height(n.left.left) >= height(n.left.right)) n = rotateRight(n); else n = rotateLeftThenRight(n); } else if (n.balance == 2) { if (height(n.right.right) >= height(n.right.left)) n = rotateLeft(n); else n = rotateRightThenLeft(n); } if (n.parent != null) { rebalance(n.parent); } else { root = n; } } private Node rotateLeft(Node a) { Node b = a.right; b.parent = a.parent; a.right = b.left; if (a.right != null) a.right.parent = a; b.left = a; a.parent = b; if (b.parent != null) { if (b.parent.right == a) { b.parent.right = b; } else { b.parent.left = b; } } setBalance(a, b); return b; } private Node rotateRight(Node a) { Node b = a.left; b.parent = a.parent; a.left = b.right; if (a.left != null) a.left.parent = a; b.right = a; a.parent = b; if (b.parent != null) { if (b.parent.right == a) { b.parent.right = b; } else { b.parent.left = b; } } setBalance(a, b); return b; } private Node rotateLeftThenRight(Node n) { n.left = rotateLeft(n.left); return rotateRight(n); } private Node rotateRightThenLeft(Node n) { n.right = rotateRight(n.right); return rotateLeft(n); } private int height(Node n) { if (n == null) return -1; return n.height; } private void setBalance(Node... nodes) { for (Node n : nodes) { reheight(n); n.balance = height(n.right) - height(n.left); } } public void printBalance() { printBalance(root); } private void printBalance(Node n) { if (n != null) { printBalance(n.left); System.out.printf("%s ", n.balance); printBalance(n.right); } } private void reheight(Node node) { if (node != null) { node.height = 1 + Math.max(height(node.left), height(node.right)); } } public static void main(String[] args) { AVLtree tree = new AVLtree(); System.out.println("Inserting values 1 to 10"); for (int i = 1; i < 10; i++) tree.insert(i); System.out.print("Printing balance: "); tree.printBalance(); }} Inserting values 1 to 10 Printing balance: 0 0 0 1 0 1 0 0 0 ### More elaborate version ## Kotlin Translation of: Java class AvlTree { private var root: Node? = null private class Node(var key: Int, var parent: Node?) { var balance: Int = 0 var left : Node? = null var right: Node? = null } fun insert(key: Int): Boolean { if (root == null) root = Node(key, null) else { var n: Node? = root var parent: Node while (true) { if (n!!.key == key) return false parent = n val goLeft = n.key > key n = if (goLeft) n.left else n.right if (n == null) { if (goLeft) parent.left = Node(key, parent) else parent.right = Node(key, parent) rebalance(parent) break } } } return true } fun delete(delKey: Int) { if (root == null) return var n: Node? = root var parent: Node? = root var delNode: Node? = null var child: Node? = root while (child != null) { parent = n n = child child = if (delKey >= n.key) n.right else n.left if (delKey == n.key) delNode = n } if (delNode != null) { delNode.key = n!!.key child = if (n.left != null) n.left else n.right if (0 == root!!.key.compareTo(delKey)) { root = child if (null != root) { root!!.parent = null } } else { if (parent!!.left == n) parent.left = child else parent.right = child if (null != child) { child.parent = parent } rebalance(parent) } } private fun rebalance(n: Node) { setBalance(n) var nn = n if (nn.balance == -2) if (height(nn.left!!.left) >= height(nn.left!!.right)) nn = rotateRight(nn) else nn = rotateLeftThenRight(nn) else if (nn.balance == 2) if (height(nn.right!!.right) >= height(nn.right!!.left)) nn = rotateLeft(nn) else nn = rotateRightThenLeft(nn) if (nn.parent != null) rebalance(nn.parent!!) else root = nn } private fun rotateLeft(a: Node): Node { val b: Node? = a.right b!!.parent = a.parent a.right = b.left if (a.right != null) a.right!!.parent = a b.left = a a.parent = b if (b.parent != null) { if (b.parent!!.right == a) b.parent!!.right = b else b.parent!!.left = b } setBalance(a, b) return b } private fun rotateRight(a: Node): Node { val b: Node? = a.left b!!.parent = a.parent a.left = b.right if (a.left != null) a.left!!.parent = a b.right = a a.parent = b if (b.parent != null) { if (b.parent!!.right == a) b.parent!!.right = b else b.parent!!.left = b } setBalance(a, b) return b } private fun rotateLeftThenRight(n: Node): Node { n.left = rotateLeft(n.left!!) return rotateRight(n) } private fun rotateRightThenLeft(n: Node): Node { n.right = rotateRight(n.right!!) return rotateLeft(n) } private fun height(n: Node?): Int { if (n == null) return -1 return 1 + Math.max(height(n.left), height(n.right)) } private fun setBalance(vararg nodes: Node) { for (n in nodes) n.balance = height(n.right) - height(n.left) } fun printKey() { printKey(root) println() } private fun printKey(n: Node?) { if (n != null) { printKey(n.left) print("${n.key} ")            printKey(n.right)        }    }     fun printBalance() {        printBalance(root)        println()    }     private fun printBalance(n: Node?) {        if (n != null) {            printBalance(n.left)            print("${n.balance} ") printBalance(n.right) } }} fun main(args: Array<String>) { val tree = AvlTree() println("Inserting values 1 to 10") for (i in 1..10) tree.insert(i) print("Printing key : ") tree.printKey() print("Printing balance : ") tree.printBalance()} Output: Inserting values 1 to 10 Printing key : 1 2 3 4 5 6 7 8 9 10 Printing balance : 0 0 0 1 0 0 0 0 1 0  ## Lua AVL={balance=0}AVL.__mt={__index = AVL} function AVL:new(list) local o={} setmetatable(o, AVL.__mt) for _,v in ipairs(list or {}) do o=o:insert(v) end return oend function AVL:rebalance() local rotated=false if self.balance>1 then if self.right.balance<0 then self.right, self.right.left.right, self.right.left = self.right.left, self.right, self.right.left.right self.right.right.balance=self.right.balance>-1 and 0 or 1 self.right.balance=self.right.balance>0 and 2 or 1 end self, self.right.left, self.right = self.right, self, self.right.left self.left.balance=1-self.balance self.balance=self.balance==0 and -1 or 0 rotated=true elseif self.balance<-1 then if self.left.balance>0 then self.left, self.left.right.left, self.left.right = self.left.right, self.left, self.left.right.left self.left.left.balance=self.left.balance<1 and 0 or -1 self.left.balance=self.left.balance<0 and -2 or -1 end self, self.left.right, self.left = self.left, self, self.left.right self.right.balance=-1-self.balance self.balance=self.balance==0 and 1 or 0 rotated=true end return self,rotatedend function AVL:insert(v) if not self.value then self.value=v self.balance=0 return self,1 end local grow if v==self.value then return self,0 elseif v<self.value then if not self.left then self.left=self:new() end self.left,grow=self.left:insert(v) self.balance=self.balance-grow else if not self.right then self.right=self:new() end self.right,grow=self.right:insert(v) self.balance=self.balance+grow end self,rotated=self:rebalance() return self, (rotated or self.balance==0) and 0 or grow end function AVL:delete_move(dir,other,mul) if self[dir] then local sb2,v self[dir], sb2, v=self[dir]:delete_move(dir,other,mul) self.balance=self.balance+sb2*mul self,sb2=self:rebalance() return self,(sb2 or self.balance==0) and -1 or 0,v else return self[other],-1,self.value endend function AVL:delete(v,isSubtree) local grow=0 if v==self.value then local v if self.balance>0 then self.right,grow,v=self.right:delete_move("left","right",-1) elseif self.left then self.left,grow,v=self.left:delete_move("right","left",1) grow=-grow else return not isSubtree and AVL:new(),-1 end self.value=v self.balance=self.balance+grow elseif v<self.value and self.left then self.left,grow=self.left:delete(v,true) self.balance=self.balance-grow elseif v>self.value and self.right then self.right,grow=self.right:delete(v,true) self.balance=self.balance+grow else return self,0 end self,rotated=self:rebalance() return self, grow~=0 and (rotated or self.balance==0) and -1 or 0end -- output functions function AVL:toList(list) if not self.value then return {} end list=list or {} if self.left then self.left:toList(list) end list[#list+1]=self.value if self.right then self.right:toList(list) end return listend function AVL:dump(depth) if not self.value then return end depth=depth or 0 if self.right then self.right:dump(depth+1) end print(string.rep(" ",depth)..self.value.." ("..self.balance..")") if self.left then self.left:dump(depth+1) endend -- test local test=AVL:new{1,10,5,15,20,3,5,14,7,13,2,8,3,4,5,10,9,8,7} test:dump()print("\ninsert 17:")test=test:insert(17)test:dump()print("\ndelete 10:")test=test:delete(10)test:dump()print("\nlist:")print(unpack(test:toList()))  Output:  20 (0) 15 (1) 14 (1) 13 (0) 10 (-1) 9 (0) 8 (0) 7 (0) 5 (-1) 4 (0) 3 (1) 2 (1) 1 (0) insert 17: 20 (0) 17 (0) 15 (0) 14 (1) 13 (0) 10 (-1) 9 (0) 8 (0) 7 (0) 5 (-1) 4 (0) 3 (1) 2 (1) 1 (0) delete 10: 20 (0) 17 (0) 15 (0) 14 (1) 13 (0) 9 (-1) 8 (-1) 7 (0) 5 (-1) 4 (0) 3 (1) 2 (1) 1 (0) list: 1 2 3 4 5 7 8 9 13 14 15 17 20 ## Objeck Translation of: Java class AVLNode { @key : Int; @balance : Int; @height : Int; @left : AVLNode; @right : AVLNode; @above : AVLNode; New(key : Int, above : AVLNode) { @key := key; @above := above; } method : public : GetKey() ~ Int { return @key; } method : public : GetLeft() ~ AVLNode { return @left; } method : public : GetRight() ~ AVLNode { return @right; } method : public : GetAbove() ~ AVLNode { return @above; } method : public : GetBalance() ~ Int { return @balance; } method : public : GetHeight() ~ Int { return @height; } method : public : SetBalance(balance : Int) ~ Nil { @balance := balance; } method : public : SetHeight(height : Int) ~ Nil { @height := height; } method : public : SetAbove(above : AVLNode) ~ Nil { @above := above; } method : public : SetLeft(left : AVLNode) ~ Nil { @left := left; } method : public : SetRight(right : AVLNode) ~ Nil { @right := right; } method : public : SetKey(key : Int) ~ Nil { @key := key; }} class AVLTree { @root : AVLNode; New() {} method : public : Insert(key : Int) ~ Bool { if(@root = Nil) { @root := AVLNode->New( key, Nil); return true; }; n := @root; while(true) { if(n->GetKey() = key) { return false; }; parent := n; goLeft := n->GetKey() > key; n := goLeft ? n->GetLeft() : n->GetRight(); if(n = Nil) { if(goLeft) { parent->SetLeft(AVLNode->New( key, parent)); } else { parent->SetRight(AVLNode->New( key, parent)); }; Rebalance(parent); break; }; }; return true; } method : Delete(node : AVLNode) ~ Nil { if (node->GetLeft() = Nil & node->GetRight() = Nil) { if (node ->GetAbove() = Nil) { @root := Nil; } else { parent := node ->GetAbove(); if (parent->GetLeft() = node) { parent->SetLeft(Nil); } else { parent->SetRight(Nil); }; Rebalance(parent); }; return; }; if (node->GetLeft() <> Nil) { child := node->GetLeft(); while (child->GetRight() <> Nil) { child := child->GetRight(); }; node->SetKey(child->GetKey()); Delete(child); } else { child := node->GetRight(); while (child->GetLeft() <> Nil) { child := child->GetLeft(); }; node->SetKey(child->GetKey()); Delete(child); }; } method : public : Delete(delKey : Int) ~ Nil { if (@root = Nil) { return; }; child := @root; while (child <> Nil) { node := child; child := delKey >= node->GetKey() ? node->GetRight() : node->GetLeft(); if (delKey = node->GetKey()) { Delete(node); return; }; }; } method : Rebalance(n : AVLNode) ~ Nil { SetBalance(n); if (n->GetBalance() = -2) { if (Height(n->GetLeft()->GetLeft()) >= Height(n->GetLeft()->GetRight())) { n := RotateRight(n); } else { n := RotateLeftThenRight(n); }; } else if (n->GetBalance() = 2) { if(Height(n->GetRight()->GetRight()) >= Height(n->GetRight()->GetLeft())) { n := RotateLeft(n); } else { n := RotateRightThenLeft(n); }; }; if(n->GetAbove() <> Nil) { Rebalance(n->GetAbove()); } else { @root := n; }; } method : RotateLeft(a : AVLNode) ~ AVLNode { b := a->GetRight(); b->SetAbove(a->GetAbove()); a->SetRight(b->GetLeft()); if(a->GetRight() <> Nil) { a->GetRight()->SetAbove(a); }; b->SetLeft(a); a->SetAbove(b); if (b->GetAbove() <> Nil) { if (b->GetAbove()->GetRight() = a) { b->GetAbove()->SetRight(b); } else { b->GetAbove()->SetLeft(b); }; }; SetBalance(a); SetBalance(b); return b; } method : RotateRight(a : AVLNode) ~ AVLNode { b := a->GetLeft(); b->SetAbove(a->GetAbove()); a->SetLeft(b->GetRight()); if (a->GetLeft() <> Nil) { a->GetLeft()->SetAbove(a); }; b->SetRight(a); a->SetAbove(b); if (b->GetAbove() <> Nil) { if (b->GetAbove()->GetRight() = a) { b->GetAbove()->SetRight(b); } else { b->GetAbove()->SetLeft(b); }; }; SetBalance(a); SetBalance(b); return b; } method : RotateLeftThenRight(n : AVLNode) ~ AVLNode { n->SetLeft(RotateLeft(n->GetLeft())); return RotateRight(n); } method : RotateRightThenLeft(n : AVLNode) ~ AVLNode { n->SetRight(RotateRight(n->GetRight())); return RotateLeft(n); } method : SetBalance(n : AVLNode) ~ Nil { Reheight(n); n->SetBalance(Height(n->GetRight()) - Height(n->GetLeft())); } method : Reheight(node : AVLNode) ~ Nil { if(node <> Nil) { node->SetHeight(1 + Int->Max(Height(node->GetLeft()), Height(node->GetRight()))); }; } method : Height(n : AVLNode) ~ Int { if(n = Nil) { return -1; }; return n->GetHeight(); } method : public : PrintBalance() ~ Nil { PrintBalance(@root); } method : PrintBalance(n : AVLNode) ~ Nil { if (n <> Nil) { PrintBalance(n->GetLeft()); balance := n->GetBalance(); "{$balance} "->Print();      PrintBalance(n->GetRight());    };  }} class Test {  function : Main(args : String[]) ~ Nil {    tree := AVLTree->New();     "Inserting values 1 to 10"->PrintLine();    for(i := 1; i < 10; i+=1;) {      tree->Insert(i);    };     "Printing balance: "->Print();    tree->PrintBalance();  }} 
Output:
Inserting values 1 to 10
Printing balance: 0 0 0 1 0 1 0 0 0


## Objective-C

Translation of: Java
 This example is incomplete. It is missing an @interface for AVLTree and also missing any @interface or @implementation for AVLTreeNode. Please ensure that it meets all task requirements and remove this message.
 @implementation AVLTree -(BOOL)insertWithKey:(NSInteger)key {     if (self.root == nil) {        self.root = [[AVLTreeNode alloc]initWithKey:key andParent:nil];    } else {         AVLTreeNode *n = self.root;        AVLTreeNode *parent;         while (true) {             if (n.key == key) {                return false;            }             parent = n;             BOOL goLeft = n.key > key;            n = goLeft ? n.left : n.right;             if (n == nil) {                 if (goLeft) {                    parent.left = [[AVLTreeNode alloc]initWithKey:key andParent:parent];                } else {                    parent.right = [[AVLTreeNode alloc]initWithKey:key andParent:parent];                }                [self rebalanceStartingAtNode:parent];                break;            }        }    }     return true;} -(void)rebalanceStartingAtNode:(AVLTreeNode*)n {     [self setBalance:@[n]];     if (n.balance == -2) {        if ([self height:(n.left.left)] >= [self height:n.left.right]) {            n = [self rotateRight:n];        } else {            n = [self rotateLeftThenRight:n];        }    } else if (n.balance == 2) {        if ([self height:n.right.right] >= [self height:n.right.left]) {            n = [self rotateLeft:n];        } else {            n = [self rotateRightThenLeft:n];        }    }     if (n.parent != nil) {        [self rebalanceStartingAtNode:n.parent];    } else {        self.root = n;    }}  -(AVLTreeNode*)rotateRight:(AVLTreeNode*)a {     AVLTreeNode *b = a.left;    b.parent = a.parent;     a.left = b.right;     if (a.left != nil) {        a.left.parent = a;    }     b.right = a;    a.parent = b;     if (b.parent != nil) {        if (b.parent.right == a) {            b.parent.right = b;        } else {            b.parent.left = b;        }    }     [self setBalance:@[a,b]];    return b; } -(AVLTreeNode*)rotateLeftThenRight:(AVLTreeNode*)n {     n.left = [self rotateLeft:n.left];    return [self rotateRight:n]; } -(AVLTreeNode*)rotateRightThenLeft:(AVLTreeNode*)n {     n.right = [self rotateRight:n.right];    return [self rotateLeft:n];} -(AVLTreeNode*)rotateLeft:(AVLTreeNode*)a {     //set a's right node as b    AVLTreeNode* b = a.right;    //set b's parent as a's parent (which could be nil)    b.parent = a.parent;    //in case b had a left child transfer it to a    a.right = b.left;     // after changing a's reference to the right child, make sure the parent is set too    if (a.right != nil) {        a.right.parent = a;    }     // switch a over to the left to be b's left child    b.left = a;    a.parent = b;     if (b.parent != nil) {        if (b.parent.right == a) {            b.parent.right = b;        } else {            b.parent.right = b;        }    }     [self setBalance:@[a,b]];     return b; }   -(void) setBalance:(NSArray*)nodesArray {     for (AVLTreeNode* n in nodesArray) {         n.balance = [self height:n.right] - [self height:n.left];    } } -(int)height:(AVLTreeNode*)n {     if (n == nil) {        return -1;    }     return 1 + MAX([self height:n.left], [self height:n.right]);} -(void)printKey:(AVLTreeNode*)n {    if (n != nil) {        [self printKey:n.left];        NSLog(@"%ld", n.key);        [self printKey:n.right];    }} -(void)printBalance:(AVLTreeNode*)n {    if (n != nil) {        [self printBalance:n.left];        NSLog(@"%ld", n.balance);        [self printBalance:n.right];    }}@end-- test  int main(int argc, const char * argv[]) {    @autoreleasepool {         AVLTree *tree = [AVLTree new];        NSLog(@"inserting values 1 to 6");        [tree insertWithKey:1];        [tree insertWithKey:2];        [tree insertWithKey:3];        [tree insertWithKey:4];        [tree insertWithKey:5];        [tree insertWithKey:6];         NSLog(@"printing balance: ");        [tree printBalance:tree.root];         NSLog(@"printing key: ");        [tree printKey:tree.root];    }    return 0;}  
Output:
inserting values 1 to 6
printing balance:
0
0
0
0
1
0

printing key:
1
2
3
4
5
6


## Phix

Translated from the C version at http://www.geeksforgeeks.org/avl-tree-set-2-deletion
The standard distribution includes demo\rosetta\AVL_tree.exw, which contains a slightly longer but perhaps more readable version, with a command line equivalent of https://www.cs.usfca.edu/~galles/visualization/AVLtree.html as well as a simple tree structure display routine and additional verification code (both modelled on the C version found on this page)

enum KEY = 0,     LEFT,     HEIGHT,    -- (NB +/-1 gives LEFT or RIGHT)     RIGHT sequence tree = {}integer freelist = 0 function newNode(object key)integer node    if freelist=0 then        node = length(tree)+1        tree &= {key,NULL,1,NULL}    else        node = freelist        freelist = tree[freelist]        tree[node+KEY..node+RIGHT] = {key,NULL,1,NULL}    end if    return nodeend function function height(integer node)    return iff(node=NULL?0:tree[node+HEIGHT])end function procedure setHeight(integer node)    tree[node+HEIGHT] = max(height(tree[node+LEFT]), height(tree[node+RIGHT]))+1end procedure function rotate(integer node, integer direction)integer idirection = LEFT+RIGHT-directioninteger pivot = tree[node+idirection]    {tree[pivot+direction],tree[node+idirection]} = {node,tree[pivot+direction]}    setHeight(node)    setHeight(pivot)    return pivotend function function getBalance(integer N)    return iff(N==NULL ? 0 : height(tree[N+LEFT])-height(tree[N+RIGHT]))end function function insertNode(integer node, object key)    if node==NULL then        return newNode(key)    end if    integer c = compare(key,tree[node+KEY])    if c!=0 then        integer direction = HEIGHT+c    -- LEFT or RIGHT        tree[node+direction] = insertNode(tree[node+direction], key)        setHeight(node)        integer balance = trunc(getBalance(node)/2) -- +/-1 (or 0)        if balance then            direction = HEIGHT-balance  -- LEFT or RIGHT            c = compare(key,tree[tree[node+direction]+KEY])            if c=balance then                tree[node+direction] = rotate(tree[node+direction],direction)            end if            if c!=0 then                node = rotate(node,LEFT+RIGHT-direction)            end if        end if    end if    return nodeend function function minValueNode(integer node)    while 1 do        integer next = tree[node+LEFT]        if next=NULL then exit end if        node = next    end while    return nodeend function function deleteNode(integer root, object key)integer c    if root=NULL then return root end if    c = compare(key,tree[root+KEY])    if c=-1 then        tree[root+LEFT] = deleteNode(tree[root+LEFT], key)    elsif c=+1 then        tree[root+RIGHT] = deleteNode(tree[root+RIGHT], key)    elsif tree[root+LEFT]==NULL       or tree[root+RIGHT]==NULL then        integer temp = iff(tree[root+LEFT] ? tree[root+LEFT] : tree[root+RIGHT])        if temp==NULL then  -- No child case            {temp,root} = {root,NULL}        else                -- One child case            tree[root+KEY..root+RIGHT] = tree[temp+KEY..temp+RIGHT]        end if        tree[temp+KEY] = freelist        freelist = temp    else                    -- Two child case        integer temp = minValueNode(tree[root+RIGHT])        tree[root+KEY] = tree[temp+KEY]        tree[root+RIGHT] = deleteNode(tree[root+RIGHT], tree[temp+KEY])    end if    if root=NULL then return root end if    setHeight(root)    integer balance = trunc(getBalance(root)/2)    if balance then        integer direction = HEIGHT-balance        c = compare(getBalance(tree[root+direction]),0)        if c=-balance then            tree[root+direction] = rotate(tree[root+direction],direction)        end if        root = rotate(root,LEFT+RIGHT-direction)    end if    return rootend function procedure inOrder(integer node)    if node!=NULL then        inOrder(tree[node+LEFT])        printf(1, "%d ", tree[node+KEY])        inOrder(tree[node+RIGHT])    end ifend procedure integer root = NULLsequence test = shuffle(tagset(50003))    for i=1 to length(test) do        root = insertNode(root,test[i])    end for    test = shuffle(tagset(50000))    for i=1 to length(test) do        root = deleteNode(root,test[i])    end for    inOrder(root)
Output:
50001 50002 50003

## Python

 """ Python AVL tree example based on  https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-introduction-to-algorithms-fall-2011/lecture-videos/lec06_code.zip Simplified for Rosetta Code example. See also: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-introduction-to-algorithms-fall-2011/lecture-videos/MIT6_006F11_lec06_orig.pdf https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-introduction-to-algorithms-fall-2011/lecture-videos/lecture-6-avl-trees-avl-sort/ """ class AVLNode(object):    """A node in the AVL tree."""     def __init__(self, parent, k):        """Creates a node.         Args:            parent: The node's parent.            k: key of the node.        """        self.key = k        self.parent = parent        self.left = None        self.right = None     def _str(self):        """Internal method for ASCII art."""        label = str(self.key)        if self.left is None:            left_lines, left_pos, left_width = [], 0, 0        else:            left_lines, left_pos, left_width = self.left._str()        if self.right is None:            right_lines, right_pos, right_width = [], 0, 0        else:            right_lines, right_pos, right_width = self.right._str()        middle = max(right_pos + left_width - left_pos + 1, len(label), 2)        pos = left_pos + middle // 2        width = left_pos + middle + right_width - right_pos        while len(left_lines) < len(right_lines):            left_lines.append(' ' * left_width)        while len(right_lines) < len(left_lines):            right_lines.append(' ' * right_width)        if (middle - len(label)) % 2 == 1 and self.parent is not None and \           self is self.parent.left and len(label) < middle:            label += '.'        label = label.center(middle, '.')        if label[0] == '.': label = ' ' + label[1:]        if label[-1] == '.': label = label[:-1] + ' '        lines = [' ' * left_pos + label + ' ' * (right_width - right_pos),                 ' ' * left_pos + '/' + ' ' * (middle-2) +                 '\\' + ' ' * (right_width - right_pos)] + \          [left_line + ' ' * (width - left_width - right_width) + right_line           for left_line, right_line in zip(left_lines, right_lines)]        return lines, pos, width     def __str__(self):        return '\n'.join(self._str()[0])     def find(self, k):        """Finds and returns the node with key k from the subtree rooted at this         node.         Args:            k: The key of the node we want to find.         Returns:            The node with key k.        """        if k == self.key:            return self        elif k < self.key:            if self.left is None:                return None            else:                return self.left.find(k)        else:            if self.right is None:                  return None            else:                return self.right.find(k)     def find_min(self):        """Finds the node with the minimum key in the subtree rooted at this         node.         Returns:            The node with the minimum key.        """        current = self        while current.left is not None:            current = current.left        return current     def next_larger(self):        """Returns the node with the next larger key (the successor) in the BST.        """        if self.right is not None:            return self.right.find_min()        current = self        while current.parent is not None and current is current.parent.right:            current = current.parent        return current.parent     def insert(self, node):        """Inserts a node into the subtree rooted at this node.         Args:            node: The node to be inserted.        """        if node is None:            return        if node.key < self.key:            if self.left is None:                node.parent = self                self.left = node            else:                self.left.insert(node)        else:            if self.right is None:                node.parent = self                self.right = node            else:                self.right.insert(node)     def delete(self):        """Deletes and returns this node from the tree."""        if self.left is None or self.right is None:            if self is self.parent.left:                self.parent.left = self.left or self.right                if self.parent.left is not None:                    self.parent.left.parent = self.parent            else:                self.parent.right = self.left or self.right                if self.parent.right is not None:                    self.parent.right.parent = self.parent            return self        else:            s = self.next_larger()            self.key, s.key = s.key, self.key            return s.delete() def height(node):    if node is None:        return -1    else:        return node.height def update_height(node):    node.height = max(height(node.left), height(node.right)) + 1 class AVL(object):    """    AVL binary search tree implementation.    """     def __init__(self):        """ empty tree """        self.root = None     def __str__(self):        if self.root is None: return '<empty tree>'        return str(self.root)     def find(self, k):        """Finds and returns the node with key k from the subtree rooted at this         node.         Args:            k: The key of the node we want to find.         Returns:            The node with key k or None if the tree is empty.        """        return self.root and self.root.find(k)     def find_min(self):        """Returns the minimum node of this BST."""         return self.root and self.root.find_min()     def next_larger(self, k):        """Returns the node that contains the next larger (the successor) key in        the BST in relation to the node with key k.         Args:            k: The key of the node of which the successor is to be found.         Returns:            The successor node.        """        node = self.find(k)        return node and node.next_larger()        def left_rotate(self, x):        y = x.right        y.parent = x.parent        if y.parent is None:            self.root = y        else:            if y.parent.left is x:                y.parent.left = y            elif y.parent.right is x:                y.parent.right = y        x.right = y.left        if x.right is not None:            x.right.parent = x        y.left = x        x.parent = y        update_height(x)        update_height(y)     def right_rotate(self, x):        y = x.left        y.parent = x.parent        if y.parent is None:            self.root = y        else:            if y.parent.left is x:                y.parent.left = y            elif y.parent.right is x:                y.parent.right = y        x.left = y.right        if x.left is not None:            x.left.parent = x        y.right = x        x.parent = y        update_height(x)        update_height(y)     def rebalance(self, node):        while node is not None:            update_height(node)            if height(node.left) >= 2 + height(node.right):                if height(node.left.left) >= height(node.left.right):                    self.right_rotate(node)                else:                    self.left_rotate(node.left)                    self.right_rotate(node)            elif height(node.right) >= 2 + height(node.left):                if height(node.right.right) >= height(node.right.left):                    self.left_rotate(node)                else:                    self.right_rotate(node.right)                    self.left_rotate(node)            node = node.parent     def insert(self, k):        """Inserts a node with key k into the subtree rooted at this node.        This AVL version guarantees the balance property: h = O(lg n).         Args:            k: The key of the node to be inserted.        """        node = AVLNode(None, k)        if self.root is None:            # The root's parent is None.            self.root = node        else:            self.root.insert(node)        self.rebalance(node)     def delete(self, k):        """Deletes and returns a node with key k if it exists from the BST.        This AVL version guarantees the balance property: h = O(lg n).         Args:            k: The key of the node that we want to delete.         Returns:            The deleted node with key k.        """        node = self.find(k)        if node is None:            return None        if node is self.root:            pseudoroot = AVLNode(None, 0)            pseudoroot.left = self.root            self.root.parent = pseudoroot            deleted = self.root.delete()            self.root = pseudoroot.left            if self.root is not None:                self.root.parent = None        else:            deleted = node.delete()           ## node.parent is actually the old parent of the node,        ## which is the first potentially out-of-balance node.        self.rebalance(deleted.parent) def test(args=None):    import random, sys    if not args:        args = sys.argv[1:]    if not args:        print('usage: %s <number-of-random-items | item item item ...>' % \              sys.argv[0])        sys.exit()    elif len(args) == 1:        items = (random.randrange(100) for i in range(int(args[0])))    else:        items = [int(i) for i in args]     tree = AVL()    print(tree)    for item in items:        tree.insert(item)        print()        print(tree) if __name__ == '__main__': test() 
Output:
python avlrc.py 1 2 3 4 5 6 7 8 9 10

... only showing last tree ...

..4...
/      \
2      .8.
/ \    /   \
1  3    6   9
/\ /\  / \  /\
5  7   10
/\ /\  /\



## Rust

See AVL tree/Rust.

## Scala

import scala.collection.mutable class AVLTree[A](implicit val ordering: Ordering[A]) extends mutable.SortedSet[A] {   if (ordering eq null) throw new NullPointerException("ordering must not be null")   private var _root: AVLNode = _  private var _size = 0   override def size: Int = _size   override def foreach[U](f: A => U): Unit = {    val stack = mutable.Stack[AVLNode]()    var current = root    var done = false     while (!done) {      if (current != null) {        stack.push(current)        current = current.left      } else if (stack.nonEmpty) {        current = stack.pop()        f.apply(current.key)         current = current.right      } else {        done = true      }    }  }   def root: AVLNode = _root   override def isEmpty: Boolean = root == null   override def min[B >: A](implicit cmp: Ordering[B]): A = minNode().key   def minNode(): AVLNode = {    if (root == null) throw new UnsupportedOperationException("empty tree")    var node = root    while (node.left != null) node = node.left    node  }   override def max[B >: A](implicit cmp: Ordering[B]): A = maxNode().key   def maxNode(): AVLNode = {    if (root == null) throw new UnsupportedOperationException("empty tree")    var node = root    while (node.right != null) node = node.right    node  }   def next(node: AVLNode): Option[AVLNode] = {    var successor = node    if (successor != null) {      if (successor.right != null) {        successor = successor.right        while (successor != null && successor.left != null) {          successor = successor.left        }      } else {        successor = node.parent        var n = node        while (successor != null && successor.right == n) {          n = successor          successor = successor.parent        }      }    }    Option(successor)  }   def prev(node: AVLNode): Option[AVLNode] = {    var predecessor = node    if (predecessor != null) {      if (predecessor.left != null) {        predecessor = predecessor.left        while (predecessor != null && predecessor.right != null) {          predecessor = predecessor.right        }      } else {        predecessor = node.parent        var n = node        while (predecessor != null && predecessor.left == n) {          n = predecessor          predecessor = predecessor.parent        }      }    }    Option(predecessor)  }   override def rangeImpl(from: Option[A], until: Option[A]): mutable.SortedSet[A] = ???   override def +=(key: A): AVLTree.this.type = {    insert(key)    this  }   def insert(key: A): AVLNode = {    if (root == null) {      _root = new AVLNode(key)      _size += 1      return root    }     var node = root    var parent: AVLNode = null    var cmp = 0     while (node != null) {      parent = node      cmp = ordering.compare(key, node.key)      if (cmp == 0) return node // duplicate      node = node.matchNextChild(cmp)    }     val newNode = new AVLNode(key, parent)    if (cmp <= 0) parent._left = newNode    else parent._right = newNode     while (parent != null) {      cmp = ordering.compare(parent.key, key)      if (cmp < 0) parent.balanceFactor -= 1      else parent.balanceFactor += 1       parent = parent.balanceFactor match {        case -1 | 1 => parent.parent        case x if x < -1 =>          if (parent.right.balanceFactor == 1) rotateRight(parent.right)          val newRoot = rotateLeft(parent)          if (parent == root) _root = newRoot          null        case x if x > 1 =>          if (parent.left.balanceFactor == -1) rotateLeft(parent.left)          val newRoot = rotateRight(parent)          if (parent == root) _root = newRoot          null        case _ => null      }    }     _size += 1    newNode  }   override def -=(key: A): AVLTree.this.type = {    remove(key)    this  }   override def remove(key: A): Boolean = {    var node = findNode(key).orNull    if (node == null) return false     if (node.left != null) {      var max = node.left       while (max.left != null || max.right != null) {        while (max.right != null) max = max.right         node._key = max.key        if (max.left != null) {          node = max          max = max.left        }      }      node._key = max.key      node = max    }     if (node.right != null) {      var min = node.right       while (min.left != null || min.right != null) {        while (min.left != null) min = min.left         node._key = min.key        if (min.right != null) {          node = min          min = min.right        }      }      node._key = min.key      node = min    }     var current = node    var parent = node.parent    while (parent != null) {      parent.balanceFactor += (if (parent.left == current) -1 else 1)       current = parent.balanceFactor match {        case x if x < -1 =>          if (parent.right.balanceFactor == 1) rotateRight(parent.right)          val newRoot = rotateLeft(parent)          if (parent == root) _root = newRoot          newRoot        case x if x > 1 =>          if (parent.left.balanceFactor == -1) rotateLeft(parent.left)          val newRoot = rotateRight(parent)          if (parent == root) _root = newRoot          newRoot        case _ => parent      }       parent = current.balanceFactor match {        case -1 | 1 => null        case _ => current.parent      }    }     if (node.parent != null) {      if (node.parent.left == node) {        node.parent._left = null      } else {        node.parent._right = null      }    }     if (node == root) _root = null     _size -= 1    true  }   def findNode(key: A): Option[AVLNode] = {    var node = root    while (node != null) {      val cmp = ordering.compare(key, node.key)      if (cmp == 0) return Some(node)      node = node.matchNextChild(cmp)    }    None  }   private def rotateLeft(node: AVLNode): AVLNode = {    val rightNode = node.right    node._right = rightNode.left    if (node.right != null) node.right._parent = node     rightNode._parent = node.parent    if (rightNode.parent != null) {      if (rightNode.parent.left == node) {        rightNode.parent._left = rightNode      } else {        rightNode.parent._right = rightNode      }    }     node._parent = rightNode    rightNode._left = node     node.balanceFactor += 1    if (rightNode.balanceFactor < 0) {      node.balanceFactor -= rightNode.balanceFactor    }     rightNode.balanceFactor += 1    if (node.balanceFactor > 0) {      rightNode.balanceFactor += node.balanceFactor    }    rightNode  }   private def rotateRight(node: AVLNode): AVLNode = {    val leftNode = node.left    node._left = leftNode.right    if (node.left != null) node.left._parent = node     leftNode._parent = node.parent    if (leftNode.parent != null) {      if (leftNode.parent.left == node) {        leftNode.parent._left = leftNode      } else {        leftNode.parent._right = leftNode      }    }     node._parent = leftNode    leftNode._right = node     node.balanceFactor -= 1    if (leftNode.balanceFactor > 0) {      node.balanceFactor -= leftNode.balanceFactor    }     leftNode.balanceFactor -= 1    if (node.balanceFactor < 0) {      leftNode.balanceFactor += node.balanceFactor    }    leftNode  }   override def contains(elem: A): Boolean = findNode(elem).isDefined   override def iterator: Iterator[A] = ???   override def keysIteratorFrom(start: A): Iterator[A] = ???   class AVLNode private[AVLTree](k: A, p: AVLNode = null) {     private[AVLTree] var _key: A = k    private[AVLTree] var _parent: AVLNode = p    private[AVLTree] var _left: AVLNode = _    private[AVLTree] var _right: AVLNode = _    private[AVLTree] var balanceFactor: Int = 0     def parent: AVLNode = _parent     private[AVLTree] def selectNextChild(key: A): AVLNode = matchNextChild(ordering.compare(key, this.key))     def key: A = _key     private[AVLTree] def matchNextChild(cmp: Int): AVLNode = cmp match {      case x if x < 0 => left      case x if x > 0 => right      case _ => null    }     def left: AVLNode = _left     def right: AVLNode = _right  } }

## Sidef

Translation of: D
class AVLtree {     has root = nil     struct Node {        Number key,        Number balance = 0,        Node left = nil,        Node right = nil,        Node parent = nil,    }     method insert(key) {        if (root == nil) {            root = Node(key)            return true        }         var n = root        var parent = nil         loop {            if (n.key == key) {                return false            }            parent = n            var goLeft = (n.key > key)            n = (goLeft ? n.left : n.right)             if (n == nil) {                var tn = Node(key, parent: parent)                if (goLeft) {                    parent.left = tn                }                else {                    parent.right = tn                }                self.rebalance(parent)                break            }        }         return true    }     method delete_key(delKey) {        if (root == nil) { return nil }         var n = root        var parent = root        var delNode = nil        var child = root         while (child != nil) {            parent = n            n = child            child = (delKey >= n.key ? n.right : n.left)            if (delKey == n.key) {                delNode = n            }        }         if (delNode != nil) {            delNode.key = n.key            child = (n.left != nil ? n.left : n.right)             if (root.key == delKey) {                root = child            }            else {                if (parent.left == n) {                    parent.left = child                }                else {                    parent.right = child                }                self.rebalance(parent)            }        }    }     method rebalance(n) {        if (n == nil) { return nil }        self.setBalance(n)         given (n.balance) {            when (-2) {                if (self.height(n.left.left) >= self.height(n.left.right)) {                    n = self.rotate(n, :right)                }                else {                    n = self.rotate_twice(n, :left, :right)                }            }            when (2) {                if (self.height(n.right.right) >= self.height(n.right.left)) {                    n = self.rotate(n, :left)                }                else {                    n = self.rotate_twice(n, :right, :left)                }            }        }         if (n.parent != nil) {            self.rebalance(n.parent)        }        else {            root = n        }    }     method rotate(a, dir) {        var b = (dir == :left ? a.right : a.left)        b.parent = a.parent         (dir == :left) ? (a.right = b.left)                       : (a.left  = b.right)         if (a.right != nil) {            a.right.parent = a        }         b.$dir = a a.parent = b if (b.parent != nil) { if (b.parent.right == a) { b.parent.right = b } else { b.parent.left = b } } self.setBalance(a, b) return b } method rotate_twice(n, dir1, dir2) { n.left = self.rotate(n.left, dir1) self.rotate(n, dir2) } method height(n) { if (n == nil) { return -1 } 1 + Math.max(self.height(n.left), self.height(n.right)) } method setBalance(*nodes) { nodes.each { |n| n.balance = (self.height(n.right) - self.height(n.left)) } } method printBalance { self.printBalance(root) } method printBalance(n) { if (n != nil) { self.printBalance(n.left) print(n.balance, ' ') self.printBalance(n.right) } }} var tree = AVLtree() say "Inserting values 1 to 10"{|i| tree.insert(i) } << 1..10print "Printing balance: "tree.printBalance Output: Inserting values 1 to 10 Printing balance: 0 0 0 1 0 0 0 0 1 0  ## Simula CLASS AVL;BEGIN ! AVL TREE ADAPTED FROM JULIENNE WALKER'S PRESENTATION AT ; ! HTTP://ETERNALLYCONFUZZLED.COM/TUTS/DATASTRUCTURES/JSW_TUT_AVL.ASPX. ; ! THIS PORT USES SIMILAR INDENTIFIER NAMES. ; ! THE KEY INTERFACE MUST BE SUPPORTED BY DATA STORED IN THE AVL TREE. ; CLASS KEY; VIRTUAL: PROCEDURE LESS IS BOOLEAN PROCEDURE LESS (K); REF(KEY) K;; PROCEDURE EQUAL IS BOOLEAN PROCEDURE EQUAL(K); REF(KEY) K;; BEGIN END KEY; ! NODE IS A NODE IN AN AVL TREE. ; CLASS NODE(DATA); REF(KEY) DATA; ! ANYTHING COMPARABLE WITH LESS AND EQUAL. ; BEGIN INTEGER BALANCE; ! BALANCE FACTOR ; REF(NODE) ARRAY LINK(0:1); ! CHILDREN, INDEXED BY "DIRECTION", 0 OR 1. ; END NODE; ! A LITTLE READABILITY FUNCTION FOR RETURNING THE OPPOSITE OF A DIRECTION, ; ! WHERE A DIRECTION IS 0 OR 1. ; ! WHERE JW WRITES !DIR, THIS CODE HAS OPP(DIR). ; INTEGER PROCEDURE OPP(DIR); INTEGER DIR; BEGIN OPP := 1 - DIR; END OPP; ! SINGLE ROTATION ; REF(NODE) PROCEDURE SINGLE(ROOT, DIR); REF(NODE) ROOT; INTEGER DIR; BEGIN REF(NODE) SAVE; SAVE :- ROOT.LINK(OPP(DIR)); ROOT.LINK(OPP(DIR)) :- SAVE.LINK(DIR); SAVE.LINK(DIR) :- ROOT; SINGLE :- SAVE; END SINGLE; ! DOUBLE ROTATION ; REF(NODE) PROCEDURE DOUBLE(ROOT, DIR); REF(NODE) ROOT; INTEGER DIR; BEGIN REF(NODE) SAVE; SAVE :- ROOT.LINK(OPP(DIR)).LINK(DIR); ROOT.LINK(OPP(DIR)).LINK(DIR) :- SAVE.LINK(OPP(DIR)); SAVE.LINK(OPP(DIR)) :- ROOT.LINK(OPP(DIR)); ROOT.LINK(OPP(DIR)) :- SAVE; SAVE :- ROOT.LINK(OPP(DIR)); ROOT.LINK(OPP(DIR)) :- SAVE.LINK(DIR); SAVE.LINK(DIR) :- ROOT; DOUBLE :- SAVE; END DOUBLE; ! ADJUST BALANCE FACTORS AFTER DOUBLE ROTATION ; PROCEDURE ADJUSTBALANCE(ROOT, DIR, BAL); REF(NODE) ROOT; INTEGER DIR, BAL; BEGIN REF(NODE) N, NN; N :- ROOT.LINK(DIR); NN :- N.LINK(OPP(DIR)); IF NN.BALANCE = 0 THEN BEGIN ROOT.BALANCE := 0; N.BALANCE := 0; END ELSE IF NN.BALANCE = BAL THEN BEGIN ROOT.BALANCE := -BAL; N.BALANCE := 0; END ELSE BEGIN ROOT.BALANCE := 0; N.BALANCE := BAL; END; NN.BALANCE := 0; END ADJUSTBALANCE; REF(NODE) PROCEDURE INSERTBALANCE(ROOT, DIR); REF(NODE) ROOT; INTEGER DIR; BEGIN REF(NODE) N; INTEGER BAL; N :- ROOT.LINK(DIR); BAL := 2*DIR - 1; IF N.BALANCE = BAL THEN BEGIN ROOT.BALANCE := 0; N.BALANCE := 0; INSERTBALANCE :- SINGLE(ROOT, OPP(DIR)); END ELSE BEGIN ADJUSTBALANCE(ROOT, DIR, BAL); INSERTBALANCE :- DOUBLE(ROOT, OPP(DIR)); END; END INSERTBALANCE; CLASS TUPLE(N,B); REF(NODE) N; BOOLEAN B;; REF(TUPLE) PROCEDURE INSERTR(ROOT, DATA); REF(NODE) ROOT; REF(KEY) DATA; BEGIN IF ROOT == NONE THEN INSERTR :- NEW TUPLE(NEW NODE(DATA), FALSE) ELSE BEGIN REF(TUPLE) T; BOOLEAN DONE; INTEGER DIR; DIR := 0; IF ROOT.DATA.LESS(DATA) THEN DIR := 1; T :- INSERTR(ROOT.LINK(DIR), DATA); ROOT.LINK(DIR) :- T.N; DONE := T.B; IF DONE THEN INSERTR :- NEW TUPLE(ROOT, TRUE) ELSE BEGIN ROOT.BALANCE := ROOT.BALANCE + 2*DIR - 1; IF ROOT.BALANCE = 0 THEN INSERTR :- NEW TUPLE(ROOT, TRUE) ELSE IF ROOT.BALANCE = 1 OR ROOT.BALANCE = -1 THEN INSERTR :- NEW TUPLE(ROOT, FALSE) ELSE INSERTR :- NEW TUPLE(INSERTBALANCE(ROOT, DIR), TRUE); END; END; END INSERTR; ! INSERT A NODE INTO THE AVL TREE. ; ! DATA IS INSERTED EVEN IF OTHER DATA WITH THE SAME KEY ALREADY EXISTS. ; PROCEDURE INSERT(TREE, DATA); NAME TREE; REF(NODE) TREE; REF(KEY) DATA; BEGIN REF(TUPLE) T; T :- INSERTR(TREE, DATA); TREE :- T.N; END INSERT; REF(TUPLE) PROCEDURE REMOVEBALANCE(ROOT, DIR); REF(NODE) ROOT; INTEGER DIR; BEGIN REF(NODE) N; INTEGER BAL; N :- ROOT.LINK(OPP(DIR)); BAL := 2*DIR - 1; IF N.BALANCE = -BAL THEN BEGIN ROOT.BALANCE := 0; N.BALANCE := 0; REMOVEBALANCE :- NEW TUPLE(SINGLE(ROOT, DIR), FALSE); END ELSE IF N.BALANCE = BAL THEN BEGIN ADJUSTBALANCE(ROOT, OPP(DIR), -BAL); REMOVEBALANCE :- NEW TUPLE(DOUBLE(ROOT, DIR), FALSE); END ELSE BEGIN ROOT.BALANCE := -BAL; N.BALANCE := BAL; REMOVEBALANCE :- NEW TUPLE(SINGLE(ROOT, DIR), TRUE); END END REMOVEBALANCE; REF(TUPLE) PROCEDURE REMOVER(ROOT, DATA); REF(NODE) ROOT; REF(KEY) DATA; BEGIN INTEGER DIR; BOOLEAN DONE; REF(TUPLE) T; IF ROOT == NONE THEN REMOVER :- NEW TUPLE(NONE, FALSE) ELSE IF ROOT.DATA.EQUAL(DATA) THEN BEGIN IF ROOT.LINK(0) == NONE THEN BEGIN REMOVER :- NEW TUPLE(ROOT.LINK(1), FALSE); GOTO L; END ELSE IF ROOT.LINK(1) == NONE THEN BEGIN REMOVER :- NEW TUPLE(ROOT.LINK(0), FALSE); GOTO L; END ELSE BEGIN REF(NODE) HEIR; HEIR :- ROOT.LINK(0); WHILE HEIR.LINK(1) =/= NONE DO HEIR :- HEIR.LINK(1); ROOT.DATA :- HEIR.DATA; DATA :- HEIR.DATA; END; END; DIR := 0; IF ROOT.DATA.LESS(DATA) THEN DIR := 1; T :- REMOVER(ROOT.LINK(DIR), DATA); ROOT.LINK(DIR) :- T.N; DONE := T.B; IF DONE THEN BEGIN REMOVER :- NEW TUPLE(ROOT, TRUE); GOTO L; END; ROOT.BALANCE := ROOT.BALANCE + 1 - 2*DIR; IF ROOT.BALANCE = 1 OR ROOT.BALANCE = -1 THEN REMOVER :- NEW TUPLE(ROOT, TRUE) ELSE IF ROOT.BALANCE = 0 THEN REMOVER :- NEW TUPLE(ROOT, FALSE) ELSE REMOVER :- REMOVEBALANCE(ROOT, DIR); L: END REMOVER; ! REMOVE A SINGLE ITEM FROM AN AVL TREE. ; ! IF KEY DOES NOT EXIST, FUNCTION HAS NO EFFECT. ; PROCEDURE REMOVE(TREE, DATA); NAME TREE; REF(NODE) TREE; REF(KEY) DATA; BEGIN REF(TUPLE) T; T :- REMOVER(TREE, DATA); TREE :- T.N; END REMOVEM; END. A demonstration program: EXTERNAL CLASS AVL; AVLBEGIN KEY CLASS INTEGERKEY(I); INTEGER I; BEGIN BOOLEAN PROCEDURE LESS (K); REF(KEY) K; LESS := I < K QUA INTEGERKEY.I; BOOLEAN PROCEDURE EQUAL(K); REF(KEY) K; EQUAL := I = K QUA INTEGERKEY.I; END INTEGERKEY; PROCEDURE DUMP(ROOT); REF(NODE) ROOT; BEGIN IF ROOT =/= NONE THEN BEGIN DUMP(ROOT.LINK(0)); OUTINT(ROOT.DATA QUA INTEGERKEY.I, 0); OUTTEXT(" "); DUMP(ROOT.LINK(1)); END END DUMP; INTEGER I; REF(NODE) TREE; OUTTEXT("Empty tree: "); DUMP(TREE); OUTIMAGE; FOR I := 3, 1, 4, 1, 5 DO BEGIN OUTTEXT("Insert "); OUTINT(I, 0); OUTTEXT(": "); INSERT(TREE, NEW INTEGERKEY(I)); DUMP(TREE); OUTIMAGE; END; FOR I := 3, 1 DO BEGIN OUTTEXT("Remove "); OUTINT(I, 0); OUTTEXT(": "); REMOVE(TREE, NEW INTEGERKEY(I)); DUMP(TREE); OUTIMAGE; END; END. Output: Empty tree: Insert 3: 3 Insert 1: 1 3 Insert 4: 1 3 4 Insert 1: 1 1 3 4 Insert 5: 1 1 3 4 5 Remove 3: 1 1 4 5 Remove 1: 1 4 5  ## Tcl Note that in general, you would not normally write a tree directly in Tcl when writing code that required an ${\displaystyle \alpha }$=${\displaystyle \rightarrow \beta }$ map, but would rather use either an array variable or a dictionary value (which are internally implemented using a high-performance hash table engine). Works with: Tcl version 8.6 package require TclOO namespace eval AVL { # Class for the overall tree; manages real public API oo::class create Tree { variable root nil class constructor {{nodeClass AVL::Node}} { set class [oo::class create Node [list superclass$nodeClass]] 	    # Create a nil instance to act as a leaf sentinel	    set nil [my NewNode ""]	    set root [$nil ref] # Make nil be special oo::objdefine$nil {		method height {} {return 0}		method key {} {error "no key possible"}		method value {} {error "no value possible"}		method destroy {} {		    # Do nothing (doesn't prohibit destruction entirely)		}		method print {indent increment} {		    # Do nothing		}	    }	} 	# How to actually manufacture a new node	method NewNode {key} {	    if {![info exists nil]} {set nil ""}	    $class new$key $nil [list [namespace current]::my NewNode] } # Create a new node in the tree and return it method insert {key} { set node [my NewNode$key]	    if {$root eq$nil} {		set root $node } else {$root insert $node } return$node	} 	# Find the node for a particular key	method lookup {key} {	    for {set node $root} {$node ne $nil} {} { if {[$node key] == $key} { return$node		} elseif {[$node key] >$key} {		    set node [$node left] } else { set node [$node right]		}	    }	    error "no such node"	} 	# Print a tree out, one node per line	method print {{indent 0} {increment 1}} {	    $root print$indent $increment return } } # Class of an individual node; may be subclassed oo::class create Node { variable key value left right 0 refcount newNode constructor {n nil instanceFactory} { set newNode$instanceFactory	    set 0 [expr {$nil eq "" ? [self] :$nil}]	    set key $n set value {} set left [set right$0]	    set refcount 0	}	method ref {} {	    incr refcount	    return [self]	}	method destroy {} {	    if {[incr refcount -1] < 1} next	}	method New {key value} {	    set n [{*}$newNode$key]	    $n setValue$value	    return $n } # Getters method key {} {return$key}	method value {} {return $value} method left {} {return$left}	method right {args} {return $right} # Setters method setValue {newValue} { set value$newValue	}	method setLeft {node} {	    # Non-trivial because of reference management	    $node ref$left destroy	    set left $node return } method setRight {node} { # Non-trivial because of reference management$node ref	    $right destroy set right$node	    return	} 	# Print a node and its descendents	method print {indent increment} {	    puts [format "%s%s => %s" [string repeat " " $indent]$key $value] incr indent$increment	    $left print$indent $increment$right print $indent$increment	} 	method height {} {	    return [expr {max([$left height], [$right height]) + 1}]	}	method balanceFactor {} {	    expr {[$left height] - [$right height]}	} 	method insert {node} {	    # Simple insertion	    if {$key > [$node key]} {		if {$left eq$0} {		    my setLeft $node } else {$left insert $node } } else { if {$right eq $0} { my setRight$node		} else {		    $right insert$node		}	    } 	    # Rebalance this node	    if {[my balanceFactor] > 1} {		if {[$left balanceFactor] < 0} {$left rotateLeft		}		my rotateRight	    } elseif {[my balanceFactor] < -1} {		if {[$right balanceFactor] > 0} {$right rotateRight		}		my rotateLeft	    }	} 	# AVL Rotations	method rotateLeft {} {	    set new [my New $key$value]	    set key [$right key] set value [$right value]	    $new setLeft$left	    $new setRight [$right left]	    my setLeft $new my setRight [$right right]	} 	method rotateRight {} {	    set new [my New $key$value]	    set key [$left key] set value [$left value]	    $new setLeft [$left right]	    $new setRight$right	    my setLeft [$left left] my setRight$new	}    }}

Demonstrating:

# Create an AVL treeAVL::Tree create tree # Populate it with some semi-random datafor {set i 33} {$i < 127} {incr i} { [tree insert$i] setValue \	[string repeat [format %c $i] [expr {1+int(rand()*5)}]]} # Print it outtree print # Look up a few values in the treefor {set i 0} {$i < 10} {incr i} {    set k [expr {33+int((127-33)*rand())}]    puts $k=>[[tree lookup$k] value]} # Destroy the tree and all its nodestree destroy
Output:
64 => @@@
48 => 000
40 => (((((