AVL tree/C sharp
Documentation
See I# for a description of the code.
Code
<lang csharp> // Finite Ordered Sets - 4State - Balanced
using System; using System.Collections.Generic;
public enum Direction { FromLeft, FromRight };
public enum State { Header, LeftHigh, Balanced, RightHigh };
public enum SetOperation {
Union, Intersection, SymmetricDifference, Difference, Equality, Inequality, Subset, Superset
}
public class Node {
public Node Left; public Node Right; public Node Parent; public State Balance;
public Node() { Left = this; Right = this; Parent = null; Balance = State.Header; }
public Node(Node p) { Left = null; Right = null; Parent = p; Balance = State.Balanced; } public bool IsHeader { get { return Balance == State.Header; } }
}
public class SetNode<T> : Node {
public T Data;
public SetNode() { } public SetNode(T dataType, Node Parent) : base(Parent) { Data = dataType; }
public override int GetHashCode() { return Data.GetHashCode(); }
}
class Utility // Nongeneric Tree Balancing {
static void RotateLeft(ref Node Root) { Node Parent = Root.Parent; Node x = Root.Right;
Root.Parent = x; x.Parent = Parent; if (x.Left != null) x.Left.Parent = Root;
Root.Right = x.Left; x.Left = Root; Root = x; }
static void RotateRight(ref Node Root) { Node Parent = Root.Parent; Node x = Root.Left;
Root.Parent = x; x.Parent = Parent; if (x.Right != null) x.Right.Parent = Root;
Root.Left = x.Right; x.Right = Root; Root = x; }
static void BalanceLeft(ref Node Root) { Node Left = Root.Left;
switch (Left.Balance) { case State.LeftHigh: Root.Balance = State.Balanced; Left.Balance = State.Balanced; RotateRight(ref Root); break;
case State.RightHigh: { Node subRight = Left.Right; switch (subRight.Balance) { case State.Balanced: Root.Balance = State.Balanced; Left.Balance = State.Balanced; break;
case State.RightHigh: Root.Balance = State.Balanced; Left.Balance = State.LeftHigh; break;
case State.LeftHigh: Root.Balance = State.RightHigh; Left.Balance = State.Balanced; break; } subRight.Balance = State.Balanced; RotateLeft(ref Left); Root.Left = Left; RotateRight(ref Root); } break;
case State.Balanced: Root.Balance = State.LeftHigh; Left.Balance = State.RightHigh; RotateRight(ref Root); break; } }
static void BalanceRight(ref Node Root) { Node Right = Root.Right;
switch (Right.Balance) { case State.RightHigh: Root.Balance = State.Balanced; Right.Balance = State.Balanced; RotateLeft(ref Root); break;
case State.LeftHigh: { Node subLeft = Right.Left; // Left Subtree of Right switch (subLeft.Balance) { case State.Balanced: Root.Balance = State.Balanced; Right.Balance = State.Balanced; break;
case State.LeftHigh: Root.Balance = State.Balanced; Right.Balance = State.RightHigh; break;
case State.RightHigh: Root.Balance = State.LeftHigh; Right.Balance = State.Balanced; break; } subLeft.Balance = State.Balanced; RotateRight(ref Right); Root.Right = Right; RotateLeft(ref Root); } break;
case State.Balanced: Root.Balance = State.RightHigh; Right.Balance = State.LeftHigh; RotateLeft(ref Root); break; } }
public static void BalanceSet(Node Root, Direction From) { bool Taller = true;
while (Taller) { Node Parent = Root.Parent; Direction NextFrom = (Parent.Left == Root) ? Direction.FromLeft : Direction.FromRight;
if (From == Direction.FromLeft) { switch (Root.Balance) { case State.LeftHigh: if (Parent.IsHeader) BalanceLeft(ref Parent.Parent); else if (Parent.Left == Root) BalanceLeft(ref Parent.Left); else BalanceLeft(ref Parent.Right); Taller = false; break;
case State.Balanced: Root.Balance = State.LeftHigh; Taller = true; break;
case State.RightHigh: Root.Balance = State.Balanced; Taller = false; break; } } else { switch (Root.Balance) { case State.LeftHigh: Root.Balance = State.Balanced; Taller = false; break;
case State.Balanced: Root.Balance = State.RightHigh; Taller = true; break;
case State.RightHigh: if (Parent.IsHeader) BalanceRight(ref Parent.Parent); else if (Parent.Left == Root) BalanceRight(ref Parent.Left); else BalanceRight(ref Parent.Right); Taller = false; break; } }
if (Taller) // skip up a level { if (Parent.IsHeader) Taller = false; else { Root = Parent; From = NextFrom; } } } }
public static void BalanceSetRemove(Node Root, Direction From) { if (Root.IsHeader) return;
bool Shorter = true;
while (Shorter) { Node Parent = Root.Parent; Direction NextFrom = (Parent.Left == Root) ? Direction.FromLeft : Direction.FromRight;
if (From == Direction.FromLeft) { switch (Root.Balance) { case State.LeftHigh: Root.Balance = State.Balanced; Shorter = true; break;
case State.Balanced: Root.Balance = State.RightHigh; Shorter = false; break;
case State.RightHigh: if (Root.Right.Balance == State.Balanced) Shorter = false; else Shorter = true; if (Parent.IsHeader) BalanceRight(ref Parent.Parent); else if (Parent.Left == Root) BalanceRight(ref Parent.Left); else BalanceRight(ref Parent.Right); break; } } else { switch (Root.Balance) { case State.RightHigh: Root.Balance = State.Balanced; Shorter = true; break;
case State.Balanced: Root.Balance = State.LeftHigh; Shorter = false; break;
case State.LeftHigh: if (Root.Left.Balance == State.Balanced) Shorter = false; else Shorter = true; if (Parent.IsHeader) BalanceLeft(ref Parent.Parent); else if (Parent.Left == Root) BalanceLeft(ref Parent.Left); else BalanceLeft(ref Parent.Right); break; } }
if (Shorter) { if (Parent.IsHeader) Shorter = false; else { From = NextFrom; Root = Parent; } } } }
public static Node PreviousItem(Node Node) { if (Node.IsHeader) { return Node.Right; }
if (Node.Left != null) { Node = Node.Left; while (Node.Right != null) Node = Node.Right; } else { Node y = Node.Parent; if (y.IsHeader) return y; while (Node == y.Left) { Node = y; y = y.Parent; } Node = y; } return Node; }
public static Node NextItem(Node Node) { if (Node.IsHeader) return Node.Left;
if (Node.Right != null) { Node = Node.Right; while (Node.Left != null) Node = Node.Left; } else { Node y = Node.Parent; if (y.IsHeader) return y; while (Node == y.Right) { Node = y; y = y.Parent; } Node = y; } return Node; }
public static ulong Depth(Node Root) { if (Root != null) { ulong Left = Root.Left != null ? Depth(Root.Left) : 0; ulong Right = Root.Right != null ? Depth(Root.Right) : 0; return Left < Right ? Right + 1 : Left + 1; } else return 0; }
static void SwapNodeReference(ref Node First, ref Node Second) { Node Temporary = First; First = Second; Second = Temporary; }
public static void SwapNodes(Node A, Node B) { if (B == A.Left) { if (B.Left != null) B.Left.Parent = A; if (B.Right != null) B.Right.Parent = A;
if (A.Right != null) A.Right.Parent = B;
if (!A.Parent.IsHeader) { if (A.Parent.Left == A) A.Parent.Left = B; else A.Parent.Right = B; } else A.Parent.Parent = B;
B.Parent = A.Parent; A.Parent = B;
A.Left = B.Left; B.Left = A;
SwapNodeReference(ref A.Right, ref B.Right); } else if (B == A.Right) { if (B.Right != null) B.Right.Parent = A; if (B.Left != null) B.Left.Parent = A;
if (A.Left != null) A.Left.Parent = B;
if (!A.Parent.IsHeader) { if (A.Parent.Left == A) A.Parent.Left = B; else A.Parent.Right = B; } else A.Parent.Parent = B;
B.Parent = A.Parent; A.Parent = B;
A.Right = B.Right; B.Right = A;
SwapNodeReference(ref A.Left, ref B.Left); } else if (A == B.Left) { if (A.Left != null) A.Left.Parent = B; if (A.Right != null) A.Right.Parent = B;
if (B.Right != null) B.Right.Parent = A;
if (!B.Parent.IsHeader) { if (B.Parent.Left == B) B.Parent.Left = A; else B.Parent.Right = A; } else B.Parent.Parent = A;
A.Parent = B.Parent; B.Parent = A;
B.Left = A.Left; A.Left = B;
SwapNodeReference(ref A.Right, ref B.Right); } else if (A == B.Right) { if (A.Right != null) A.Right.Parent = B; if (A.Left != null) A.Left.Parent = B;
if (B.Left != null) B.Left.Parent = A;
if (!B.Parent.IsHeader) { if (B.Parent.Left == B) B.Parent.Left = A; else B.Parent.Right = A; } else B.Parent.Parent = A;
A.Parent = B.Parent; B.Parent = A;
B.Right = A.Right; A.Right = B;
SwapNodeReference(ref A.Left, ref B.Left); } else { if (A.Parent == B.Parent) SwapNodeReference(ref A.Parent.Left, ref A.Parent.Right); else { if (!A.Parent.IsHeader) { if (A.Parent.Left == A) A.Parent.Left = B; else A.Parent.Right = B; } else A.Parent.Parent = B;
if (!B.Parent.IsHeader) { if (B.Parent.Left == B) B.Parent.Left = A; else B.Parent.Right = A; } else B.Parent.Parent = A; }
if (B.Left != null) B.Left.Parent = A; if (B.Right != null) B.Right.Parent = A;
if (A.Left != null) A.Left.Parent = B; if (A.Right != null) A.Right.Parent = B;
SwapNodeReference(ref A.Left, ref B.Left); SwapNodeReference(ref A.Right, ref B.Right); SwapNodeReference(ref A.Parent, ref B.Parent); }
State Balance = A.Balance; A.Balance = B.Balance; B.Balance = Balance; }
}
public struct SetEntry<T> : IEnumerator<T> {
public SetEntry(Node N) { _Node = N; }
public T Value { get { return ((SetNode<T>)_Node).Data; } }
public bool IsEnd { get { return _Node.IsHeader; } }
public bool MoveNext() { _Node = Utility.NextItem(_Node); return _Node.IsHeader ? false : true; }
public bool MovePrevious() { _Node = Utility.PreviousItem(_Node); return _Node.IsHeader ? false : true; }
public static SetEntry<T> operator ++(SetEntry<T> entry) { entry._Node = Utility.NextItem(entry._Node); return entry; }
public static SetEntry<T> operator --(SetEntry<T> entry) { entry._Node = Utility.PreviousItem(entry._Node); return entry; }
public void Reset() { while (!MoveNext()) ; }
object System.Collections.IEnumerator.Current { get { return ((SetNode<T>)_Node).Data; } }
T IEnumerator<T>.Current { get { return ((SetNode<T>)_Node).Data; } }
public static bool operator ==(SetEntry<T> x, SetEntry<T> y) { return x._Node == y._Node; } public static bool operator !=(SetEntry<T> x, SetEntry<T> y) { return x._Node != y._Node; }
public override bool Equals(object o) { return _Node == ((SetEntry<T>)o)._Node; }
public override int GetHashCode() { return _Node.GetHashCode(); }
public static SetEntry<T> operator +(SetEntry<T> C, ulong Increment) { SetEntry<T> Result = new SetEntry<T>(C._Node); for (ulong i = 0; i < Increment; i++) ++Result; return Result; }
public static SetEntry<T> operator +(ulong Increment, SetEntry<T> C) { SetEntry<T> Result = new SetEntry<T>(C._Node); for (ulong i = 0; i < Increment; i++) ++Result; return Result; }
public static SetEntry<T> operator -(SetEntry<T> C, ulong Decrement) { SetEntry<T> Result = new SetEntry<T>(C._Node); for (ulong i = 0; i < Decrement; i++) --Result; return Result; }
public override string ToString() { return Value.ToString(); }
public void Dispose() { }
public Node _Node;
}
class Set<T> : IEnumerable<T> {
IComparer<T> Comparer; Node Header; ulong Nodes;
//*** Constructors ***
public Set() { Comparer = Comparer<T>.Default; Header = new Node(); Nodes = 0; }
public Set(IComparer<T> c) { Comparer = c; Header = new Node(); Nodes = 0; }
//*** Properties ***
SetNode<T> Root { get { return (SetNode<T>)Header.Parent; } set { Header.Parent = value; } }
Node LeftMost { get { return Header.Left; } set { Header.Left = value; } }
Node RightMost { get { return Header.Right; } set { Header.Right = value; } }
public SetEntry<T> Begin { get { return new SetEntry<T>(Header.Left); } }
public SetEntry<T> End { get { return new SetEntry<T>(Header); } }
public ulong Length { get { return Nodes; } }
public ulong Depth { get { return Utility.Depth(Root); } }
//*** Operators ***
public bool this[T key] { get { return Search(key); } }
public static Set<T> operator +(Set<T> set, T t) { set.Add(t); return set; }
public static Set<T> operator -(Set<T> set, T t) { set.Remove(t); return set; }
public static Set<T> operator |(Set<T> A, Set<T> B) { Set<T> U = new Set<T>(A.Comparer); CombineSets(A, B, U, SetOperation.Union); return U; }
public static Set<T> operator &(Set<T> A, Set<T> B) { Set<T> I = new Set<T>(A.Comparer); CombineSets(A, B, I, SetOperation.Intersection); return I; }
public static Set<T> operator ^(Set<T> A, Set<T> B) { Set<T> S = new Set<T>(A.Comparer); CombineSets(A, B, S, SetOperation.SymmetricDifference); return S; }
public static Set<T> operator -(Set<T> A, Set<T> B) { Set<T> S = new Set<T>(A.Comparer); CombineSets(A, B, S, SetOperation.Difference); return S; }
public static bool operator ==(Set<T> A, Set<T> B) { return CheckSets(A, B, SetOperation.Equality); }
public static bool operator !=(Set<T> A, Set<T> B) { return CheckSets(A, B, SetOperation.Inequality); }
public override bool Equals(object o) { return CheckSets(this, (Set<T>)o, SetOperation.Equality); }
//*** Methods ***
public void Add(T key) { if (Root == null) { Root = new SetNode<T>(key, Header); LeftMost = RightMost = Root; } else { SetNode<T> Search = Root; for (; ; ) { int Compare = Comparer.Compare(key, Search.Data);
if (Compare == 0) // Item Exists throw new EntryAlreadyExistsException();
else if (Compare < 0) { if (Search.Left != null) Search = (SetNode<T>)Search.Left; else { Search.Left = new SetNode<T>(key, Search); if (LeftMost == Search) LeftMost = (SetNode<T>)Search.Left; Utility.BalanceSet(Search, Direction.FromLeft); Nodes++; } }
else { if (Search.Right != null) Search = (SetNode<T>)Search.Right; else { Search.Right = new SetNode<T>(key, Search); if (RightMost == Search) RightMost = (SetNode<T>)Search.Right; Utility.BalanceSet(Search, Direction.FromRight); Nodes++; break; } } } } }
System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator() { return new SetEntry<T>(Header); }
IEnumerator<T> IEnumerable<T>.GetEnumerator() { return new SetEntry<T>(Header); }
public override int GetHashCode() { return GetHashCode((SetNode<T>)Header.Parent); }
int GetHashCode(SetNode<T> Root) { if (Root != null) { int HashCode = Root.GetHashCode();
if (Root.Left != null) HashCode += GetHashCode((SetNode<T>)Root.Left);
if (Root.Right != null) HashCode += GetHashCode((SetNode<T>)Root.Right);
return HashCode; }
return 0; }
public void Remove(T key) { SetNode<T> root = Root;
for (; ; ) { if (root == null) throw new EntryNotFoundException();
int Compare = Comparer.Compare(key, root.Data);
if (Compare < 0) root = (SetNode<T>)root.Left;
else if (Compare > 0) root = (SetNode<T>)root.Right;
else // Item is found { if (root.Left != null && root.Right != null) { SetNode<T> replace = (SetNode<T>)root.Left; while (replace.Right != null) replace = (SetNode<T>)replace.Right; Utility.SwapNodes(root, replace); }
SetNode<T> Parent = (SetNode<T>)root.Parent;
Direction From = (Parent.Left == root) ? Direction.FromLeft : Direction.FromRight;
if (LeftMost == root) { SetEntry<T> e = new SetEntry<T>(root); e.MoveNext();
if (e._Node.IsHeader) { LeftMost = Header; RightMost = Header; } else LeftMost = e._Node; } else if (RightMost == root) { SetEntry<T> e = new SetEntry<T>(root); e.MovePrevious();
if (e._Node.IsHeader) { LeftMost = Header; RightMost = Header; } else RightMost = e._Node; }
if (root.Left == null) { if (Parent == Header) Header.Parent = root.Right; else if (Parent.Left == root) Parent.Left = root.Right; else Parent.Right = root.Right;
if (root.Right != null) root.Right.Parent = Parent; } else { if (Parent == Header) Header.Parent = root.Left; else if (Parent.Left == root) Parent.Left = root.Left; else Parent.Right = root.Left;
if (root.Left != null) root.Left.Parent = Parent; }
Utility.BalanceSetRemove(Parent, From); Nodes--; break; } } }
public bool Search(T key) { if (Root == null) return false; else { SetNode<T> Search = Root;
do { int Result = Comparer.Compare(key, Search.Data);
if (Result < 0) Search = (SetNode<T>)Search.Left;
else if (Result > 0) Search = (SetNode<T>)Search.Right;
else break;
} while (Search != null);
if (Search == null) return false; else return true; } }
public override string ToString() { string StringOut = "{";
SetEntry<T> start = Begin; SetEntry<T> end = End; SetEntry<T> last = End - 1;
while (start != end) { string new_StringOut = start.Value.ToString(); if (start != last) new_StringOut = new_StringOut + ","; StringOut = StringOut + new_StringOut; ++start; }
StringOut = StringOut + "}"; return StringOut; }
public void Validate() { if (Nodes == 0 || Root == null) { if (Nodes != 0) { throw new InvalidEmptyTreeException(); } if (Root != null) { throw new InvalidEmptyTreeException(); } if (LeftMost != Header) { throw new InvalidEndItemException(); } if (RightMost != Header) { throw new InvalidEndItemException(); } }
Validate(Root);
if (Root != null) { SetNode<T> x = Root; while (x.Left != null) x = (SetNode<T>)x.Left;
if (LeftMost != x) throw new InvalidEndItemException();
SetNode<T> y = Root; while (y.Right != null) y = (SetNode<T>)y.Right;
if (RightMost != y) throw new InvalidEndItemException(); } }
void Validate(SetNode<T> root) { if (root == null) return;
if (root.Left != null) { SetNode<T> Left = (SetNode<T>)root.Left;
if (Comparer.Compare(Left.Data, root.Data) >= 0) throw new OutOfKeyOrderException();
if (Left.Parent != root) throw new TreeInvalidParentException();
Validate((SetNode<T>)root.Left); }
if (root.Right != null) { SetNode<T> Right = (SetNode<T>)root.Right;
if (Comparer.Compare(Right.Data, root.Data) <= 0) throw new OutOfKeyOrderException();
if (Right.Parent != root) throw new TreeInvalidParentException();
Validate((SetNode<T>)root.Right); }
ulong depth_Left = root.Left != null ? Utility.Depth(root.Left) : 0; ulong depth_Right = root.Right != null ? Utility.Depth(root.Right) : 0;
if (depth_Left > depth_Right && depth_Left - depth_Right > 2) throw new TreeOutOfBalanceException();
if (depth_Left < depth_Right && depth_Right - depth_Left > 2) throw new TreeOutOfBalanceException(); }
public static void CombineSets(Set<T> A, Set<T> B, Set<T> R, SetOperation operation) { IComparer<T> TComparer = R.Comparer; SetEntry<T> First1 = A.Begin; SetEntry<T> Last1 = A.End; SetEntry<T> First2 = B.Begin; SetEntry<T> Last2 = B.End;
switch (operation) { case SetOperation.Union: while (First1 != Last1 && First2 != Last2) { int Order = TComparer.Compare(First1.Value, First2.Value);
if (Order < 0) { R.Add(First1.Value); First1.MoveNext(); }
else if (Order > 0) { R.Add(First2.Value); First2.MoveNext(); }
else { R.Add(First1.Value); First1.MoveNext(); First2.MoveNext(); } } while (First1 != Last1) { R.Add(First1.Value); First1.MoveNext(); } while (First2 != Last2) { R.Add(First2.Value); First2.MoveNext(); } return;
case SetOperation.Intersection: while (First1 != Last1 && First2 != Last2) { int Order = TComparer.Compare(First1.Value, First2.Value);
if (Order < 0) First1.MoveNext();
else if (Order > 0) First2.MoveNext();
else { R.Add(First1.Value); First1.MoveNext(); First2.MoveNext(); } } return;
case SetOperation.SymmetricDifference: while (First1 != Last1 && First2 != Last2) { int Order = TComparer.Compare(First1.Value, First2.Value);
if (Order < 0) { R.Add(First1.Value); First1.MoveNext(); }
else if (Order > 0) { R.Add(First2.Value); First2.MoveNext(); }
else { First1.MoveNext(); First2.MoveNext(); } }
while (First1 != Last1) { R.Add(First1.Value); First1.MoveNext(); }
while (First2 != Last2) { R.Add(First2.Value); First2.MoveNext(); } return;
case SetOperation.Difference: while (First1 != Last1 && First2 != Last2) { int Order = TComparer.Compare(First1.Value, First2.Value);
if (Order < 0) { R.Add(First1.Value); First1.MoveNext(); }
else if (Order > 0) { R.Add(First1.Value); First1.MoveNext(); First2.MoveNext(); }
else { First1.MoveNext(); First2.MoveNext(); } }
while (First1 != Last1) { R.Add(First1.Value); First1.MoveNext(); } return; }
throw new InvalidSetOperationException(); }
public static bool CheckSets(Set<T> A, Set<T> B, SetOperation operation) { IComparer<T> TComparer = A.Comparer; SetEntry<T> First1 = A.Begin; SetEntry<T> Last1 = A.End; SetEntry<T> First2 = B.Begin; SetEntry<T> Last2 = B.End;
switch (operation) { case SetOperation.Equality: case SetOperation.Inequality: { bool Equals = true;
while (First1 != Last1 && First2 != Last2) { if (TComparer.Compare(First1.Value, First2.Value) == 0) { First1.MoveNext(); First2.MoveNext(); } else { Equals = false; break; } }
if (Equals) { if (First1 != Last1) Equals = false; if (First2 != Last2) Equals = false; }
if (operation == SetOperation.Equality) return Equals; else return !Equals; }
case SetOperation.Subset: case SetOperation.Superset: { bool Subset = true;
while (First1 != Last1 && First2 != Last2) { int Order = TComparer.Compare(First1.Value, First2.Value);
if (Order < 0) { Subset = false; break; }
else if (Order > 0) First2.MoveNext();
else { First1.MoveNext(); First2.MoveNext(); } }
if (Subset) if (First1 != Last1) Subset = false;
if (operation == SetOperation.Subset) return Subset; else return !Subset; } }
throw new InvalidSetOperationException(); }
}
public class EntryNotFoundException : Exception {
static String message = "The requested entry could not be located in the specified collection.";
public EntryNotFoundException() : base(message) { }
}
public class EntryAlreadyExistsException : Exception {
static String message = "The requested entry already resides in the collection.";
public EntryAlreadyExistsException() : base(message) { }
}
public class InvalidEndItemException : Exception {
static String message = "The validation routines detected that the end item of a tree is invalid.";
public InvalidEndItemException() : base(message) { }
}
public class InvalidEmptyTreeException : Exception {
static String message = "The validation routines detected that an empty tree is invalid.";
public InvalidEmptyTreeException() : base(message) { }
}
public class OutOfKeyOrderException : Exception {
static String message = "A trees was found to be out of key order.";
public OutOfKeyOrderException() : base(message) { }
}
public class TreeInvalidParentException : Exception {
static String message = "The validation routines detected that the Parent structure of a tree is invalid.";
public TreeInvalidParentException() : base(message) { }
}
public class TreeOutOfBalanceException : Exception {
static String message = "The validation routines detected that the tree is out of State.";
public TreeOutOfBalanceException() : base(message) { }
}
public class InvalidSetOperationException : Exception {
static String message = "An invalid set operation was requested.";
public InvalidSetOperationException() : base(message) { }
}
class Program {
static void Main() { Set<string> s = new Set<string>() {"S0","S1","S2","S3","S4", "S5","S6","S7","S8","S9"};
Console.WriteLine("Depth = {0}", s.Depth);
s.Validate();
for (int i = 0; i < 10; i += 2) s.Remove("S" + i.ToString());
Console.WriteLine("Depth = {0}", s.Depth);
s.Validate();
Console.WriteLine("{0}", s);
Set<int> A = new Set<int>() { 1, 3, 5, 7 }; Set<int> B = new Set<int>() { 2, 4, 6, 8 };
Set<int> U = A | B;
Console.WriteLine("{0} | {1} == {2}", A, B, U); }
} </lang>