Question

PROMPT:

Consider a binary tree (not necessarily a binary search tree) with node structure.

struct Node int val; struct Node * left; struct Node* right; The following function findMax returns the largest value 'val in the tree; and returns -1 if the tree is empty. You may assume that all the values 'val' in the tree are nonnegative. struct Node * findMax(struct Node root) if (rootNULL) return -1; maxval = root->val; maxval = max(maxval, findMax(root->left); return max(maxval, findMax(root->right)); int max(int x, int y) if (x > y) return x; else return y;

QUESTION:

Prove that findMax works by mathematical induction.

Answer 1

Soln :

if the tree contains only one node,.i.e only root without left and right child then findMax returns the value in that root because that is infact the maximum value in the tree.S in case of h=1,findMax works well.

Let for h=n,findMax works well.

Here we count from the leaf nodes,h is the height of the tree.

then for h=n+1,

findMax starts at the root.it is not null.thus maxval now contains value in the root.

next maxval takes the value max(maxval,findMax(root->left)).here root->left is a BST with height h=n,so findMax works well.Lastly it returns max(maxval,findMax(root->right)).here root->right is also a BST with height h=n,so findMax works perfectly.

hence the true Maximum is returned in this case also.