Parametric polymorphism

From Rosetta Code
Parametric polymorphism
You are encouraged to solve this task according to the task description, using any language you may know.
Parametric Polymorphism is a way to define types or functions that are generic over other types. The genericity can be expressed by using type variables for the parameter type, and by a mechanism to explicitly or implicitly replace the type variables with concrete types when necessary.

Write a small example for a type declaration that is parametric over another type, together with a short bit of code (and its type signature) that uses it. A good example is a container type, let's say a binary tree, together with some function that traverses the tree, say, a map-function that operates on every element of the tree.

This language feature only applies to statically-typed languages.


type Element_Type is private;
package Container is
type Tree is tagged private;
procedure Replace_All(The_Tree : in out Tree; New_Value : Element_Type);
type Node;
type Node_Access is access Node;
type Tree tagged record
Value : Element_type;
Left  : Node_Access := null;
Right : Node_Access := null;
end record;
end Container;
package body Container is
procedure Replace_All(The_Tree : in out Tree; New_Value : Element_Type) is
The_Tree.Value := New_Value;
If The_Tree.Left /= null then
end if;
if The_tree.Right /= null then
end if;
end Replace_All;
end Container;


If the goal is to separate algorithms from types at compile type, C may do it by macros. Here's sample code implementing binary tree with node creation and insertion:
#include <stdio.h>
#include <stdlib.h>
#define decl_tree_type(T) \
typedef struct node_##T##_t node_##T##_t, *node_##T; \
struct node_##T##_t { node_##T left, right; T value; }; \
node_##T node_##T##_new(T v) { \
node_##T node = malloc(sizeof(node_##T##_t)); \
node->value = v; \
node->left = node->right = 0; \
return node; \
} \
node_##T node_##T##_insert(node_##T root, T v) { \
node_##T n = node_##T##_new(v); \
while (root) { \
if (root->value < n->value) \
if (!root->left) return root->left = n; \
else root = root->left; \
else \
if (!root->right) return root->right = n; \
else root = root->right; \
} \
return 0; \

#define tree_node(T) node_##T
#define node_insert(T, r, x) node_##T##_insert(r, x)
#define node_new(T, x) node_##T##_new(x)
int main()
int i;
tree_node(double) root_d = node_new(double, (double)rand() / RAND_MAX);
for (i = 0; i < 10000; i++)
node_insert(double, root_d, (double)rand() / RAND_MAX);
tree_node(int) root_i = node_new(int, rand());
for (i = 0; i < 10000; i++)
node_insert(int, root_i, rand());
return 0;

Comments: It's ugly looking, but it gets the job done. It has the drawback that all methods need to be re-created for each tree data type used, but hey, C++ template does that, too.

Arguably more interesting is run time polymorphism, which can't be trivially done; if you are confident in your coding skill, you could keep track of data types and method dispatch at run time yourself -- but then, you are probably too confident to not realize you might be better off using some higher level languages.


template<class T> 
class tree
T value;
tree *left;
tree *right;
void replace_all (T new_value);

For simplicity, we replace all values in the tree with a new value:

template<class T>
void tree<T>::replace_all (T new_value)
value = new_value;
if (left != NULL)
left->replace_all (new_value);
if (right != NULL)
right->replace_all (new_value);


namespace RosettaCode {
class BinaryTree<T> {
public T value;
public BinaryTree<T> left;
public BinaryTree<T> right;
public BinaryTree(T value) {
this.value = value;
public BinaryTree<U> Map<U>(Func<T,U> f) {
BinaryTree<U> Tree = new BinaryTree<U>(f(this.value));
if (left != null) {
Tree.left = left.Map(f);
if (right != null) {
Tree.right = right.Map(f);
return Tree;

Sample that creates a tree to hold int values:

namespace RosettaCode {
class Program {
static void Main(string[] args) {
BinaryTree<int> b = new BinaryTree<int>(6);
b.left = new BinaryTree<int>(5);
b.right = new BinaryTree<int>(7);
BinaryTree<double> b2 = b.Map(x => x * 10.0);


class BinaryTree<Data>(shared Data data, shared BinaryTree<Data>? left = null, shared BinaryTree<Data>? right = null) {
shared BinaryTree<NewData> myMap<NewData>(NewData f(Data d)) =>
BinaryTree {
data = f(data);
left = left?.myMap(f);
right = right?.myMap(f);
shared void run() {
value tree1 = BinaryTree {
data = 3;
left = BinaryTree {
data = 4;
right = BinaryTree {
data = 5;
left = BinaryTree {
data = 6;
value tree2 = tree1.myMap((x) => x * 333.33);


::Tree a = Empty | Node a (Tree a) (Tree a)
mapTree :: (a -> b) (Tree a) -> (Tree b)
mapTree f Empty = Empty
mapTree f (Node x l r) = Node (f x) (mapTree f l) (mapTree f r)

A digression:

Note that for the most usefulness in practical programming, a map operation like this should not be defined with a separate name but rather as fmap in an instance of the Functor type class:

instance Functor Tree where
fmap f Empty = Empty
fmap f (Node x l r) = Node (f x) (fmap f l) (fmap f r)

fmap can then be used exactly where mapTree can, but doing this also allows the use of Trees with other components which are parametric over any type which is a Functor. For example, this function will add 1 to any collection of any kind of number:

add1Everywhere :: (f a) -> (f a) | Functor f & Num a
add1Everywhere nums = fmap (\x = x + 1) nums

If we have a tree of integers, i.e. f is Tree and a is Integer, then the type of add1Everywhere is Tree Integer -> Tree Integer.

Common Lisp[edit]

Common Lisp is not statically typed, but types can be defined which are parameterized over other types. In the following piece of code, a type pair is defined which accepts two (optional) type specifiers. An object is of type (pair :car car-type :cdr cdr-type) if an only if it is a cons whose car is of type car-type and whose cdr is of type cdr-type.

(deftype pair (&key (car 't) (cdr 't))
`(cons ,car ,cdr))


> (typep (cons 1 2) '(pair :car number :cdr number))
> (typep (cons 1 2) '(pair :car number :cdr character))


class ArrayTree(T, uint N) {
T[N] data;
typeof(this) left, right;
this(T initValue) {[] = initValue; }
void tmap(const void delegate(ref typeof(data)) dg) {
if (left) left.tmap(dg);
if (right) right.tmap(dg);
void main() { // Demo code.
import std.stdio;
// Instantiate the template ArrayTree of three doubles.
alias AT3 = ArrayTree!(double, 3);
// Allocate the tree root.
auto root = new AT3(1.00);
// Add some nodes.
root.left = new AT3(1.10);
root.left.left = new AT3(1.11);
root.left.right = new AT3(1.12);
root.right = new AT3(1.20);
root.right.left = new AT3(1.21);
root.right.right = new AT3(1.22);
// Now the tree has seven nodes.
// Show the arrays of the whole tree.
//root.tmap(x => writefln("%(%.2f %)", x));
root.tmap((ref x) => writefln("%(%.2f %)", x));
// Modify the arrays of the whole tree.
//root.tmap((x){ x[] += 10; });
root.tmap((ref x){ x[] += 10; });
// Show the arrays of the whole tree again.
//root.tmap(x => writefln("%(%.2f %)", x));
root.tmap((ref x) => writefln("%(%.2f %)", x));
1.00 1.00 1.00
1.10 1.10 1.10
1.11 1.11 1.11
1.12 1.12 1.12
1.20 1.20 1.20
1.21 1.21 1.21
1.22 1.22 1.22

11.00 11.00 11.00
11.10 11.10 11.10
11.11 11.11 11.11
11.12 11.12 11.12
11.20 11.20 11.20
11.21 11.21 11.21
11.22 11.22 11.22


class TreeNode<T> {
T value;
TreeNode<T> left;
TreeNode<T> right;
TreeNode map(T f(T t)) {
var node = new TreeNode(f(value));
if(left != null) {
node.left =;
if(right != null) {
node.right =;
return node;
void forEach(void f(T t)) {
if(left != null) {
if(right != null) {
void main() {
TreeNode root = new TreeNode(1);
root.left = new TreeNode(2);
root.right = new TreeNode(3);
root.left.right = new TreeNode(4);
print('first tree');
var newRoot = => t * 222);
print('second tree');
first tree
second tree


While E itself does not do static (before evaluation) type checking, E does have guards which form a runtime type system, and has typed collections in the standard library. Here, we implement a typed tree, and a guard which accepts trees of a specific type.

(Note: Like some other examples here, this is an incomplete program in that the tree provides no way to insert or delete nodes.)

(Note: The guard definition is arguably messy boilerplate; future versions of E may provide a scheme where the interface expression can itself be used to describe parametricity, and message signatures using the type parameter, but this has not been implemented or fully designed yet. Currently, this example is more of “you can do it if you need to” than something worth doing for every data structure in your program.)

interface TreeAny guards TreeStamp {}
def Tree {
to get(Value) {
def Tree1 {
to coerce(specimen, ejector) {
def tree := TreeAny.coerce(specimen, ejector)
if (tree.valueType() != Value) {
throw.eject(ejector, "Tree value type mismatch")
return tree
return Tree1
def makeTree(T, var value :T, left :nullOk[Tree[T]], right :nullOk[Tree[T]]) {
def tree implements TreeStamp {
to valueType() { return T }
to map(f) {
value := f(value) # the declaration of value causes this to be checked
if (left != null) {
if (right != null) {
return tree
? def t := makeTree(int, 0, null, null)
# value: <tree>
? t :Tree[String]
# problem: Tree value type mismatch
? t :Tree[Int]
# problem: Failed: Undefined variable: Int
? t :Tree[int]
# value: <tree>


namespace RosettaCode
type BinaryTree<'T> =
| Element of '
| Tree of 'T * BinaryTree<'T> * BinaryTree<'T>
member this.Map(f) =
match this with
| Element(x) -> Element(f x)
| Tree(x,left,right) -> Tree((f x), left.Map(f), right.Map(f))

We can test this binary tree like so:

let t1 = Tree(2, Element(1), Tree(4,Element(3),Element(5)) )
let t2 = t1.Map(fun x -> x * 10)


Fortran does not offer polymorphism by parameter type, which is to say, enables the same source code to be declared applicable for parameters of different types, so that a contained statement such as X = A + B*C would work for any combination of integer or floating-point or complex variables as actual parameters, since exactly that (source) code would be workable in every case. Further, there is no standardised pre-processor protocol whereby one could replicate such code to produce a separate subroutine or function specific to every combination.

However, with F90 came the MODULE protocol with facilities suitable for defining "generic" subroutines or functions, or so it appears:
      MODULE SORTSEARCH		!Genuflect towards Prof. D. Knuth.
INTERFACE FIND !Binary chop search, not indexed.
1 FINDI4, !I: of integers.
2 FINDF4,FINDF8, !F: of numbers.
3 FINDTTI2,FINDTTI4 !T: of texts.
INTEGER FUNCTION FINDI4(THIS,NUMB,N) !Binary chopper. Find i such that THIS = NUMB(i)
USE ASSISTANCE !Only for the trace stuff.
INTENT(IN) THIS,NUMB,N !Imply read-only, but definitely no need for any "copy-back".
INTEGER*4 THIS,NUMB(1:*) !Where is THIS in array NUMB(1:N)?
INTEGER N !The count. In other versions, it is supplied by the index.
INTEGER L,R,P !Fingers.
Chop away.
L = 0 !Establish outer bounds.
R = N + 1 !One before, and one after, the first and last.
1 P = (R - L)/2 !Probe point offset. Beware integer overflow with (L + R)/2.
IF (P.LE.0) THEN !Aha! Nowhere! And THIS follows NUMB(L).
FINDI4 = -L !Having -L rather than 0 (or other code) might be of interest.
RETURN !Finished.
END IF !So much for exhaustion.
P = P + L !Convert from offset to probe point.
IF (THIS - NUMB(P)) 3,4,2 !Compare to the probe point.
2 L = P !Shift the left bound up: THIS follows NUMB(P).
GO TO 1 !Another chop.
3 R = P !Shift the right bound down: THIS precedes NUMB(P).
GO TO 1 !Try again.
Caught it! THIS = NUMB(P)
4 FINDI4 = P !So, THIS is found, here!
END FUNCTION FINDI4 !On success, THIS = NUMB(FINDI4); no fancy index here...

There would be a function (with a unique name) for each of the contemplated variations in parameter types, and when the compiler reached an invocation of FIND(...) it would select by matching amongst the combinations that had been defined in the routines named in the INTERFACE statement. The various actual functions could have different code, and in this case, only the INTEGER*4 THIS,NUMB(1:*) need be changed, say to REAL*4 THIS,NUMB(1:*) for FINDF4, which is why both variables are named in the one statement. However, for searching CHARACTER arrays, because the character comparison operations differ from those for numbers (and, no three-way IF-test either), additional changes are required. Thus, function FIND would appear to be a polymorphic function that accepts and returns a variety of types, but it is not, and indeed, there is actually no function called FIND anywhere in the compiled code.

That said, some systems had polymorphic variables, such as the B6700 whereby integers were represented as floating-point numbers and so exactly the same function could be presented with an integer or a floating-point variable (provided the compiler didn't check for parameter type matching - but this was routine) and it would work - so long as no divisions were involved since addition, subtraction, and multiplication are the same for both, but integer division discards any remainders. More recent computers following the Intel 8087 floating-point processor and similar add novel states to the scheme for floating-point arithmetic: not just zero and "gradual underflow" but "Infinity" and "Not a Number", which last violates even more of the axia of mathematics in that NaN does not equal NaN. In turn, this forces a modicum of polymorphism into the language so as to contend with the additional features, such as the special function IsNaN(x).

More generally, using the same code for different types of variable can be problematical. A scheme that works in single precision may not work in double precision (or vice-versa) or may not give corresponding levels of accuracy, or not converge at all, etc. While F90 also standardised special functions that give information about the precision of variables and the like, and in principle, a method could be coded that, guided by such information, would work for different precisions, this sort of scheme is beset by all manner of difficulties in problems more complex than the simple examples of text books.

Polymorphism just exacerbates the difficulties, thus, on page 219 of 16-Bit Modern Microcomputers by G. M. Corsline appears the remark "At least some of the generalized numerical solutions to common mathematical procedures have coding that is so involved and tricky in order to take care of all possible roundoff contingencies that they have been termed 'pornographic algorithms'.". And "Mathematical software is easy for the uninitiated to write but notoriously hard for the expert. This paradox exists because the beginner is satisfied if his code usually works in his own machine while the expert attempts, against overwhelming obstacles, to produce programs that always work on a large number of computers. The problem is that while standard formulas of mathematics are fairly easy to translate into FORTRAN they often are subject to instabilities due to roundoff error." - quoting John Palmer, 1980, Intel Corporation.

But sometimes it is not so troublesome, as in Pathological_floating_point_problems#The_Chaotic_Bank_Society whereby the special EPSILON(x) function that reports on the precision of a nominated variable of type x is used to determine the point beyond which further calculation (in that precision, for that formula) will make no difference.

Having flexible facilities available my lead one astray. Consider the following data aggregate, as became available with F90:
INTEGER CODE !A key number.
CHARACTER*6 NAME !Associated data.
TYPE(STUFF) TABLE(600) !An array of such entries.

Suppose the array was in sorted order by each entry's value of CODE so that TABLE(1).CODE <= TABLE(2).CODE, etc. and one wished to find the index of an entry with a specific value, x, of CODE. It is pleasing to be able to write FIND(x,TABLE.CODE,N) and have it accepted by the compiler. Rather less pleasing is that it runs very slowly.

This is because consecutive elements in an array are expected to occupy consecutive locations in storage, but the CODE elements do not, being separated by the other elements of the aggregate. So, the compiler generates code to copy the required elements to a work area, presents that as the actual parameter, and copies from the work area back on return from the function, thereby vitiating the speed advantages of the binary search. This is why the INTENT(IN) might help in such situations, as will writing FIND(x,TABLE(1:N).CODE,N) should N be often less than the full size of the table. But really, in-line code for each such usage is the only answer, despite the lack of a pre-processor to generate it.

Other options are to remain with the older-style of Fortran, using separately-defined arrays having a naming convention such as TABLECODE(600), TABLENAME(600), etc. thus not gaining the unity of declaring a TYPE, or, declaring the size within the type as in INTEGER CODE(600) except that this means that the size is a part of the type and different-sized tables would require different types, or, perhaps the compiler will handle this problem by passing a "stride" value for every array dimension so that subroutines and functions can index such parameters properly - at the cost of yet more overhead for parameter passing, and more complex indexing calculations.

In short, the available polymorphism whereby a parameter can be a normal array, or, an array-like "selection" of a component from an array of compound entities enables appealing syntax, but disasterous performance.


The parametric function in this example is the function average. It's type parameter is the interface type intCollection, and its logic uses the polymorphic function mapElements. In Go terminology, average is an ordinary function whose parameter happens to be of interface type. Code inside of average is ordinary code that just happens to call the mapElements method of its parameter. This code accesses the underlying static type only through the interface and so has no knowledge of the details of the static type or even which static type it is dealing with.

Function main creates objects t1 and t2 of two different static types, binaryTree an bTree. Both types implement the interface intCollection. t1 and t2 have different static types, but when they are passed to average, they are bound to parameter c, of interface type, and their static types are not visible within average.

Implementation of binaryTree and bTree is dummied, but you can see that implementation of average of binaryTree contains code specific to its representation (left, right) and that implementation of bTree contains code specific to its representation (buckets.)

package main
import "fmt"
func average(c intCollection) float64 {
var sum, count int
c.mapElements(func(n int) {
sum += n
return float64(sum) / float64(count)
func main() {
t1 := new(binaryTree)
t2 := new(bTree)
a1 := average(t1)
a2 := average(t2)
fmt.Println("binary tree average:", a1)
fmt.Println("b-tree average:", a2)
type intCollection interface {
type binaryTree struct {
// dummy representation details
left, right bool
func (t *binaryTree) mapElements(visit func(int)) {
// dummy implementation
if t.left == t.right {
type bTree struct {
// dummy representation details
buckets int
func (t *bTree) mapElements(visit func(int)) {
// dummy implementation
if t.buckets >= 0 {


binary tree average: 2.6666666666666665
b-tree average: 5


Translation of: Java
(more or less)


class Tree<T> {
T value
Tree<T> left
Tree<T> right
Tree(T value = null, Tree<T> left = null, Tree<T> right = null) {
this.value = value
this.left = left
this.right = right
void replaceAll(T value) {
this.value = value


data Tree a = Empty | Node a (Tree a) (Tree a)
mapTree :: (a -> b) -> Tree a -> Tree b
mapTree f Empty = Empty
mapTree f (Node x l r) = Node (f x) (mapTree f l) (mapTree f r)

A digression:

Note that for the most usefulness in practical programming, a map operation like this should not be defined with a separate name but rather as fmap in an instance of the Functor type class:

instance Functor Tree where
fmap f Empty = Empty
fmap f (Node x l r) = Node (f x) (fmap f l) (fmap f r)

fmap can then be used exactly where mapTree can, but doing this also allows the use of Trees with other components which are parametric over any type which is a Functor. For example, this function will add 1 to any collection of any kind of number:

add1Everywhere :: (Functor f, Num a) => f a -> f a
add1Everywhere nums = fmap (\x -> x + 1) nums

If we have a tree of integers, i.e. f is Tree and a is Integer, then the type of add1Everywhere is Tree Integer -> Tree Integer.

Inform 7[edit]

Phrases (the equivalent of global functions) can be defined with type parameters:

Polymorphism is a room.
To find (V - K) in (L - list of values of kind K):
repeat with N running from 1 to the number of entries in L:
if entry N in L is V:
say "Found [V] at entry [N] in [L].";
say "Did not find [V] in [L]."
When play begins:
find "needle" in {"parrot", "needle", "rutabaga"};
find 6 in {2, 3, 4};
end the story.

Inform 7 does not allow user-defined parametric types. Some built-in types can be parameterized, though:

list of numbers
relation of texts to rooms
object based rulebook producing a number
description of things
activity on things
number valued property
text valued table column
phrase (text, text) -> number

Icon and Unicon[edit]

Like PicoLisp, Icon and Unicon are dynamically typed and hence inherently polymorphic. Here's an example that can apply a function to the nodes in an n-tree regardless of the type of each node. It is up to the function to decide what to do with a given type of node. Note that the nodes do no even have to be of the same type.

procedure main()
bTree := [1, [2, [4, [7]], [5]], [3, [6, [8], [9]]]]
mapTree(bTree, write)
bTree := [1, ["two", ["four", [7]], [5]], [3, ["six", ["eight"], [9]]]]
mapTree(bTree, write)
procedure mapTree(tree, f)
every f(\tree[1]) | mapTree(!tree[2:0], f)


In J, all functions are generic over other types.

Alternatively, J is statically typed in the sense that it supports only one data type (the array), though of course inspecting a value can reveal additional details (such as: is it an array of numbers?)

(That said, note that J also supports some types which are not, strictly speaking, data. These are the verb, adverb and conjunction types. To fit this nomenclature, data is of type "noun". Also, nouns have some additional taxonomy which is beyond the scope of this task.)


Following the C++ example:

public class Tree<T>{
private T value;
private Tree<T> left;
private Tree<T> right;
public void replaceAll(T value){
this.value = value;
if(left != null)
if(right != null)


Translation of: C#
// version 1.0.6
class BinaryTree<T>(var value: T) {
var left : BinaryTree<T>? = null
var right: BinaryTree<T>? = null
fun <U> map(f: (T) -> U): BinaryTree<U> {
val tree = BinaryTree<U>(f(value))
if (left != null) tree.left = left?.map(f)
if (right != null) tree.right = right?.map(f)
return tree
fun showTopThree() = "(${left?.value}, $value, ${right?.value})"
fun main(args: Array<String>) {
val b = BinaryTree(6)
b.left = BinaryTree(5)
b.right = BinaryTree(7)
val b2 = { it * 10.0 }
(5, 6, 7)
(50.0, 60.0, 70.0)


:- type tree(A) ---> empty ; node(A, tree(A), tree(A)).
:- func map(func(A) = B, tree(A)) = tree(B).
map(_, empty) = empty.
map(F, node(A, Left, Right)) = node(F(A), map(F, Left), map(F, Right)).


type Tree[T] = ref object
value: T
left, right: Tree[T]


Translation of: C++
Works with: Xcode version 7
@interface Tree<T> : NSObject {
T value;
Tree<T> *left;
Tree<T> *right;
- (void)replaceAll:(T)v;
@implementation Tree
- (void)replaceAll:(id)v {
value = v;
[left replaceAll:v];
[right replaceAll:v];

Note that the generic type variable is only used in the declaration, but not in the implementation.


type 'a tree = Empty | Node of 'a * 'a tree * 'a tree
(** val map_tree : ('a -> 'b) -> 'a tree -> 'b tree *)
let rec map_tree f = function
| Empty -> Empty
| Node (x,l,r) -> Node (f x, map_tree f l, map_tree f r)

Perl 6[edit]

role BinaryTree[::T] {
has T $.value;
has BinaryTree[T] $.left;
has BinaryTree[T] $.right;
method replace-all(T $value) {
$!value = $value;
$!left.replace-all($value) if $!left.defined;
$!right.replace-all($value) if $!right.defined;
class IntTree does BinaryTree[Int] { }
my IntTree $it .= new(value => 1,
left => => 2),
right => => 3));
say $it.perl;
Output: => 42, left => => 42, left => BinaryTree[T], right => BinaryTree[T]), right => => 42, left => BinaryTree[T], right => BinaryTree[T]))


Phix is naturally polymorphic, with optional static typing.
The standard builtin type hierarcy is trivial:

        <-------- object --------->
        |                |
        +-atom           +-sequence
          |                |
          +-integer        +-string

User defined types are subclasses of those.
If you declare a parameter as type integer then obviously it is optimised for that, and crashes when given something else (with a clear human-readable message and file name/line number). If you declare a parameter as type object then it can handle anything you can throw at it - integers, floats, strings, or (deeply) nested sequences.
Of course many builtin routines are naturally generic, such as sort and print.
Most programming languages would throw a hissy fit if you tried to sort (or print) a mixed collection of strings and integers, but not Phix:


For comparison purposes (and because this entry looked a bit sparse without it) this is the D example from this page translated to Phix.
Note that tmap has to be a function rather than a procedure with a reference parameter, but this still achieves pass-by-reference/in-situ updates, mainly because root is a local rather than global/static, and is the target of (aka assigned to/overwritten on return from) the top-level tmap() call, and yet it would also manage the C#/Dart/Kotlin thing (by which I am referring to those specific examples on this page) of creating a whole new tree, just by changing that top-level call to object root2 = tmap(root,rid).

enum data, left, right
function tmap(sequence tree, integer rid)
tree[data] = call_func(rid,{tree[data]})
if tree[left]!=null then tree[left] = tmap(tree[left],rid) end if
if tree[right]!=null then tree[right] = tmap(tree[right],rid) end if
return tree
end function
function newnode(object v)
return {v,null,null}
end function
function add10(atom x) return x+10 end function
procedure main()
object root = newnode(1.00)
-- Add some nodes.
root[left] = newnode(1.10)
root[left][left] = newnode(1.11)
root[left][right] = newnode(1.12)
root[right] = newnode(1.20)
root[right][left] = newnode(1.21)
root[right][right] = newnode(1.22)
-- Now the tree has seven nodes.
-- Show the whole tree.
-- Modify the whole tree.
root = tmap(root,routine_id("add10"))
-- Show the whole tree again.
end procedure


PicoLisp is dynamically-typed, so in principle every function is polymetric over its arguments. It is up to the function to decide what to do with them. A function traversing a tree, modifying the nodes in-place (no matter what the type of the node is):

(de mapTree (Tree Fun)
(set Tree (Fun (car Tree)))
(and (cadr Tree) (mapTree @ Fun))
(and (cddr Tree) (mapTree @ Fun)) )


(balance 'MyTree (range 1 7))          # Create a tree of numbers
-> NIL

: (view MyTree T)                      # Display it
-> NIL

: (mapTree MyTree inc)                 # Increment all nodes
-> NIL

: (view MyTree T)                      # Display the tree
-> NIL

: (balance 'MyTree '("a" "b" "c" "d" "e" "f" "g"))  # Create a tree of strings
-> NIL

: (view MyTree T)                      # Display it
-> NIL

: (mapTree MyTree uppc)                # Convert all nodes to upper case
-> NIL

: (view MyTree T)                      # Display the tree
-> NIL


Typed Racket has parametric polymorphism:

#lang typed/racket
(define-type (Tree A) (U False (Node A)))
(struct: (A) Node
([val : A] [left : (Tree A)] [right : (Tree A)])
(: tree-map (All (A B) (A -> B) (Tree A) -> (Tree B)))
(define (tree-map f tree)
(match tree
[#f #f]
[(Node val left right)
(Node (f val) (tree-map f left) (tree-map f right))]))
;; unit tests
(require typed/rackunit)
(tree-map add1 (Node 5 (Node 3 #f #f) #f))
(Node 6 (Node 4 #f #f) #f))


This REXX programming example is modeled after the   D   example.

/*REXX program demonstrates (with displays) a method of parametric polymorphism in REXX.*/
call newRoot 1.00, 3 /*new root, and also indicate 3 stems.*/
/* [↓] no need to label the stems. */
call addStem 1.10 /*a new stem and its initial value. */
call addStem 1.11 /*" " " " " " " */
call addStem 1.12 /*" " " " " " " */
call addStem 1.20 /*" " " " " " " */
call addStem 1.21 /*" " " " " " " */
call addStem 1.22 /*" " " " " " " */
call sayNodes /*display some nicely formatted values.*/
call modRoot 50 /*modRoot will add fifty to all stems. */
call sayNodes /*display some nicely formatted values.*/
exit /*stick a fork in it, we're all done. */
addStem: nodes=nodes+1; do j=1 for stems; root.nodes.j=arg(1); end; return
modRoot: do j=1 for nodes /*traipse through all the defined nodes*/
do k=1 for stems
if datatype(root.j.k, 'N') then root.j.k=root.j.k + arg(1) /*bias.*/
end /*k*/ /* [↑] only add if numeric stem value.*/
end /*j*/
newRoot: stems=arg(2); nodes= -1 /*set NODES to a kind of "null". */
call addStem copies('═', 9); call addStem arg(1)
sayNodes: say; do j=0 to nodes; _= /*ensure each of the nodes gets shown. */
do k=1 for stems; _=_ right(root.j.k, 9)
end /*k*/
say substr(_, 2) /*ignore the first (leading) blank. */
end /*j*/
say left('', stems*11) || '('nodes" nodes)" /*also show number of nodes.*/
output   when using the default input:
═════════ ═════════ ═════════
     1.00      1.00      1.00
     1.10      1.10      1.10
     1.11      1.11      1.11
     1.12      1.12      1.12
     1.20      1.20      1.20
     1.21      1.21      1.21
     1.22      1.22      1.22
                                 (7 nodes)

═════════ ═════════ ═════════
    51.00     51.00     51.00
    51.10     51.10     51.10
    51.11     51.11     51.11
    51.12     51.12     51.12
    51.20     51.20     51.20
    51.21     51.21     51.21
    51.22     51.22     51.22
                                 (7 nodes)


struct TreeNode<T> {
value: T,
left: Option<Box<TreeNode<T>>>,
right: Option<Box<TreeNode<T>>>,
impl <T> TreeNode<T> {
fn my_map<U,F>(&self, f: &F) -> TreeNode<U> where
F: Fn(&T) -> U {
TreeNode {
value: f(&self.value),
left: match self.left {
None => None,
Some(ref n) => Some(Box::new(n.my_map(f))),
right: match self.right {
None => None,
Some(ref n) => Some(Box::new(n.my_map(f))),
fn main() {
let root = TreeNode {
value: 3,
left: Some(Box::new(TreeNode {
value: 55,
left: None,
right: None,
right: Some(Box::new(TreeNode {
value: 234,
left: Some(Box::new(TreeNode {
value: 0,
left: None,
right: None,
right: None,
root.my_map(&|x| { println!("{}" , x)});
let new_root = root.my_map(&|x| *x as f64 * 333.333f64);
new_root.my_map(&|x| { println!("{}" , x) });


There's much to be said about parametric polymorphism in Scala. Let's first see the example in question:

case class Tree[+A](value: A, left: Option[Tree[A]], right: Option[Tree[A]]) {
def map[B](f: A => B): Tree[B] =
Tree(f(value), left map (, right map (

Note that the type parameter of the class Tree, [+A]. The plus sign indicates that Tree is co-variant on A. That means Tree[X] will be a subtype of Tree[Y] if X is a subtype of Y. For example:

class Employee(val name: String)
class Manager(name: String) extends Employee(name)
val t = Tree(new Manager("PHB"), None, None)
val t2: Tree[Employee] = t

The second assignment is legal because t is of type Tree[Manager], and since Manager is a subclass of Employee, then Tree[Manager] is a subtype of Tree[Employee].

Another possible variance is the contra-variance. For instance, consider the following example:

def toName(e: Employee) =
val treeOfNames =

This works, even though map is expecting a function from Manager into something, but toName is a function of Employee into String, and Employee is a supertype, not a subtype, of Manager. It works because functions have the following definition in Scala:

trait Function1[-T1, +R]

The minus sign indicates that this trait is contra-variant in T1, which happens to be the type of the argument of the function. In other words, it tell us that, Employee => String is a subtype of Manager => String, because Employee is a supertype of Manager. While the concept of contra-variance is not intuitive, it should be clear to anyone that toName can handle arguments of type Manager, but, were not for the contra-variance, it would not be usable with a Tree[Manager].

Let's add another method to Tree to see another concept:

case class Tree[+A](value: A, left: Option[Tree[A]], right: Option[Tree[A]]) {
def map[B](f: A => B): Tree[B] =
Tree(f(value), left map (, right map (
def find[B >: A](what: B): Boolean =
(value == what) || ||

The type parameter of find is [B >: A]. That means the type is some B, as long as that B is a supertype of A. If I tried to declare what: A, Scala would not accept it. To understand why, let's consider the following code:

if (t2.find(new Employee("Dilbert")))
println("Call Catbert!")

Here we have find receiving an argument of type Employee, even though the tree it was defined on is of type Manager. The co-variance of Tree means a situation such as this is possible.

There is also an operator <:, with the opposite meaning of >:.

Finally, Scala also allows abstract types. Abtract types are similar to abstract methods: they have to be defined when a class is inherited. One simple example would be:

trait DFA {
type Element
val map = new collection.mutable.HashMap[Element, DFA]()

A concrete class wishing to inherit from DFA would need to define Element. Abstract types aren't all that different from type parameters. Mainly, they ensure that the type will be selected in the definition site (the declaration of the concrete class), and not at the usage site (instantiation of the concrete class). The difference is mainly one of style, though.


In Seed7 types like array and struct are not built-in, but are defined with parametric polymorphism. In the Seed7 documentation the terms "template" and "function with type parameters and type result" are used instead of "parametric polymorphism". E.g.: array is actually a function, which takes an element type as parameter and returns a type. To concentrate on the essentials, the example below defines the type container as array. Note that the map function has three parameters: aContainer, aVariable, and aFunc. When map is called aVariable is used also in the actual parameter of aFunc: map(container1, num, num + 1)

$ include "seed7_05.s7i";
const func type: container (in type: elemType) is func
var type: container is void;
container := array elemType;
const func container: map (in container: aContainer,
inout elemType: aVariable, ref func elemType: aFunc) is func
var container: mapResult is container.value;
for aVariable range aContainer do
mapResult &:= aFunc;
end for;
end func;
end global;
end func;
const type: intContainer is container(integer);
var intContainer: container1 is [] (1, 2, 4, 6, 10, 12, 16, 18, 22);
var intContainer: container2 is 0 times 0;
const proc: main is func
var integer: num is 0;
container2 := map(container1, num, num + 1);
for num range container2 do
write(num <& " ");
end for;
end func;


2 3 5 7 11 13 17 19 23 

Standard ML[edit]

datatype 'a tree = Empty | Node of 'a * 'a tree * 'a tree
(** val map_tree = fn : ('a -> 'b) -> 'a tree -> 'b tree *)
fun map_tree f Empty = Empty
| map_tree f (Node (x,l,r)) = Node (f x, map_tree f l, map_tree f r)


Translation of: Java
class Tree<T> {
var value: T?
var left: Tree<T>?
var right: Tree<T>?
func replaceAll(value: T?) {
self.value = value

Another version based on Algebraic Data Types:

Works with: Swift version 2+
enum Tree<T> {
case Empty
indirect case Node(T, Tree<T>, Tree<T>)
func map<U>(f : T -> U) -> Tree<U> {
switch(self) {
case .Empty  : return .Empty
case let .Node(x, l, r): return .Node(f(x),,


Types are first class entities and functions to construct or operate on them may be defined routinely. A parameterized binary tree type can be defined using a syntax for anonymous recursion in type expressions as in this example,

binary_tree_of "node-type" = "node-type"%hhhhWZAZ

or by way of a recurrence solved using a fixed point combinator imported from a library as shown below.

#import tag
#fix general_type_fixer 1
binary_tree_of "node-type" = ("node-type",(binary_tree_of "node-type")%Z)%drWZwlwAZ

(The %Z type operator constructs a "maybe" type, i.e., the free union of its operand type with the null value. Others shown above are standard stack manipulation primitives, e.g. d (dup) and w (swap), used to build the type expression tree.) At the other extreme, one may construct an equivalent parameterized type in point-free form.

binary_tree_of = %-hhhhWZAZ

A mapping combinator over this type can be defined with pattern matching like this

binary_tree_map "f" = ~&a^& ^A/"f"@an ~&amPfamPWB

or in point free form like this.

binary_tree_map = ~&a^&+ ^A\~&amPfamPWB+ @an

Here is a test program defining a type of binary trees of strings, and a function that concatenates each node with itself.

string_tree = binary_tree_of %s
x = 'foo': ('bar': (),'baz': ())
#cast string_tree
example = (binary_tree_map "s". "s"--"s") x

Type signatures are not necessarily associated with function declarations, but have uses in the other contexts such as assertions and compiler directives (e.g., #cast). Here is the output.

'foofoo': ('barbar': (),'bazbaz': ())

Visual Prolog[edit]

tree{Type} = branch(tree{Type} Left, tree{Type} Right); leaf(Type Value).
class predicates
treewalk : (tree{X},function{X,Y}) -> tree{Y} procedure (i,i).
treewalk(branch(Left,Right),Func) = branch(NewLeft,NewRight) :-
NewLeft = treewalk(Left,Func), NewRight = treewalk(Right,Func).
treewalk(leaf(Value),Func) = leaf(X) :-
X = Func(Value).
X = branch(leaf(2), branch(leaf(3),leaf(4))),
Y = treewalk(X,addone),