An Introduction to Functional Programming Through Lambda Calculus

- 182 -For example:- val root = empty;> val root = empty : inttree- val root = add 5 root;> val root = node(5,empty, empty) : inttree- val root = add 3 root;> val root = node(5,node(3,empty,empty),empty) : inttree- val root = add 7 root;> val root = node(5,node(3,empty,empty),node(7,empty,empty)) : inttree- val root = add 2 root;> val root = node(5,node(3,node(2,empty,empty),empty),node(7,empty,empty)) : inttree- val root = add 4 root;> val root = node(5,node(3,node(2,empty,empty),node(4,empty,empty)),node(7,empty,empty)) : inttree- val root = add 9 root;> val root = node(5,node(3,node(2,empty,empty),node(4,empty,empty)),node(7,empty,node(9,empty,empty))) : inttreeGiven an integer binary tree, to construct an ordered list of node values: if the tree is empty then return the empty list;otherwise, traverse the left sub-tree, pick up the root node value and traverse the right subtree:- fun traverse empty = [] |traverse (node(v:int,l:inttree,r:inttree)) =append (traverse l) (v::traverse r);> val traverse = fn : inttree -> (int list)For example:- traverse root;> [2,3,4,5,7,9] : int listWe can rewrite the above datatype to specify trees of polymorphic type by abstracting with the type variable ’a:- datatype ’a tree = empty | node of ’a * (’a tree) * (’a tree);> datatype ’a tree = empty | node of ’a * (’a tree) * (’a tree)