An Introduction to Functional Programming Through Lambda Calculus

- 139 -7.14. Binary tree efficiencyWhen an ordered binary tree is formed from a value sequence, each node holds an ordered sub-sequence. Every subnodeon the left branch of a node contains values which are less than the node’s value and every sub-node on the rightbranch contains values which are greater than the node’s value. Thus, when searching a tree for a node given a value,the selection of one branch discounts all the sub-nodes, and hence all the values, on the other branch. The number ofcomparisons required to find a node depends on how many layers of sub-nodes there are between the root and thatnode.A binary tree is said to be balanced if for any node, the number of values in both branches is the same. For abalanced binary tree, if a node holds N values then there are (N-1)/2 values in its left branch and (N-1)/2 in theits right branch. Thus, the total number of branch layers depends on how often the number of values can be halved.This suggests that in general, if:2 L ... ->[1,EMPTY,[2,EMPTY,[3,EMPTY,[4,EMPTY,EMPTY]]]]We will not consider the construction of balanced trees here.