# Number of ways to divide a Binary tree into two halves

Given a binary tree, the task is to count the number of ways to remove a single edge from the tree such that the tree gets divided into two halves with equal sum.

**Examples:**

Input:1 / \ -1 -1 \ 1Output:1 Only way to do this will be to remove the edge from the right of the root. After that we will get 2 sub-trees with sum = 0. 1 / -1 and -1 \ 1 will be the two sub-trees.Input:1 / \ -1 -1 \ -1Output:2

A **simple** solution will be to remove all the edges of the tree one by one and check if that splits the tree into two halves with the same sum. If it does, we will increase the final answer by 1. This will take O(N^{2}) time in the worst case where “N” is the number of nodes in the tree.

**Efficient appraoch:**

- Create a variable ‘sum’ and store the sum of all the elements of the Binary tree in it. We can find the sum of all the elements of a Binary tree in O(N) time as discussed in this article.
- Now we perform the following steps recursively starting from root node:
- Find the sum of all the elements of its right sub-tree (“R”). If it’s equal to half of the total sum, we increase the count by 1. This is because removing the edge connecting the current node with its right child will divide the tree into two trees with equal sum.
- Find the sum of all the elements of its left sub-tree (“L”). If it’s equal to half of the total sum, we increase the count by 1.

Below is the implementation of the above approach:

`// C++ implementation of the approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Node of a binary tree ` `struct` `node { ` ` ` `int` `data; ` ` ` `node* left; ` ` ` `node* right; ` ` ` `node(` `int` `data) ` ` ` `{ ` ` ` `this` `->data = data; ` ` ` `left = NULL; ` ` ` `right = NULL; ` ` ` `} ` `}; ` ` ` `// Function to find the sum of ` `// all the nodes of BST ` `int` `findSum(node* curr) ` `{ ` ` ` `// If current node is ` ` ` `// null ` ` ` `if` `(curr == NULL) ` ` ` `return` `0; ` ` ` ` ` `// Else ` ` ` `return` `curr->data + findSum(curr->left) ` ` ` `+ findSum(curr->right); ` `} ` ` ` `// Function to recursively check ` `// if removing any edge divides tree into ` `// two halves ` `int` `checkSum(node* curr, ` `int` `sum, ` `int` `& ans) ` `{ ` ` ` `// Variable to store the ` ` ` `// sum from left and right ` ` ` `// child ` ` ` `int` `l = 0, r = 0; ` ` ` ` ` `// Checking sum from left sub-tree ` ` ` `// if its not null ` ` ` `if` `(curr->left != NULL) { ` ` ` `l = checkSum(curr->left, sum, ans); ` ` ` `if` `(2 * l == sum) ` ` ` `ans++; ` ` ` `} ` ` ` ` ` `// Checking sum from right sub-tree ` ` ` `// if its not null ` ` ` `if` `(curr->right != NULL) { ` ` ` `r = checkSum(curr->right, sum, ans); ` ` ` `if` `(2 * r == sum) ` ` ` `ans++; ` ` ` `} ` ` ` ` ` `// Finding the sum of all the elements ` ` ` `// of current node ` ` ` `return` `l + r + curr->data; ` `} ` ` ` `// Function to return the number ` `// of ways to remove an edge ` `int` `cntWays(node* root) ` `{ ` ` ` `// If root is null ` ` ` `if` `(root == NULL) ` ` ` `return` `0; ` ` ` ` ` `// To store the final answer ` ` ` `int` `ans = 0; ` ` ` ` ` `// Sum of all the elements of BST ` ` ` `int` `sum = findSum(root); ` ` ` ` ` `// If sum is odd then it won't be possible ` ` ` `// to break it into two halves ` ` ` `if` `(sum % 2 == 1) ` ` ` `return` `0; ` ` ` ` ` `// Calling the checkSum function ` ` ` `checkSum(root, sum, ans); ` ` ` ` ` `// Returning the final answer ` ` ` `return` `ans; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `node* root = ` `new` `node(1); ` ` ` `root->left = ` `new` `node(-1); ` ` ` `root->right = ` `new` `node(-1); ` ` ` `root->right->right = ` `new` `node(1); ` ` ` ` ` `// Print the count of possible ways ` ` ` `cout << cntWays(root); ` ` ` ` ` `return` `0; ` `} ` |

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**Output:**

1

Time complexity of this approach will be O(N) and space complexity will O(H) where “N” equals number of node in Binary tree and “H” equals height of the Binary Tree.

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