## Prune nodes not on paths with given sum

Prune nodes not on paths with given sum is a very commonly asked question in Amazon interviews. It involves two concepts in one problem. First, how to find a path with a given sum and then second, how to prune nodes from binary tree.

Given a binary tree, prune nodes which are not paths with a given sum.

For example, given the below binary tree and given sum as 43, red nodes will be pruned as they are not the paths with sum 43.

### Prune nodes in a binary tree: thoughts

To solve this problem, first, understand how to find paths with a given sum in a binary tree. To prune all nodes which are not on these paths, get all the nodes which are not part of any path and then delete those nodes one by one. It requires two traversals of the binary tree.

Is it possible to delete a node while calculating the path with a given sum? At what point we find that this is not the path with given sum? At the leaf node.

Once we know that this leaf node is not part of the path with given sum, we can safely delete it. What happens to this leaf node? We directly cannot delete the parent node as there may be another subtree which leads to a path with the given sum. Hence for every node, the pruning is dependent on what comes up from its subtrees processing.

At the leaf node, we return to parent `false` if this leaf node cannot be part of the path and delete the leaf node. At parent node, we look for return values from both the subtrees. If both subtrees return `false`, it means this node is not part of the path with the given sum. If one of the subtrees returns `true`, it means the current node is part of a path with the given sum. It should not be deleted and should return `true` to its parent.

#### Prune nodes from a binary tree: implementation

#include<stdio.h> #include<stdlib.h> #include<math.h> struct node{ int value; struct node *left; struct node *right; }; typedef struct node Node; #define true 1 #define false 0 int prunePath(Node *node, int sum ){ if( !node ) return true; int subSum = sum - node->value; /* To check if left tree or right sub tree contributes to total sum */ int leftVal = false, rightVal = false; /*Check if node is leaf node */ int isLeaf = !( node->left || node->right ); /* If node is leaf node and it is part of path with sum = given sum return true to parent node so tha parent node is not deleted */ if(isLeaf && !subSum ) return true; /* If node is leaf and it not part of path with sum equals to given sum Return false to parent node */ else if(isLeaf && subSum ){ free(node); return false; } /* If node is not leaf and there is left child Traverse to left subtree*/ leftVal = prunePath(node->left, subSum); /* If node is not leaf and there is right child Traverse to right subtree*/ rightVal = prunePath(node->right, subSum); /* This is crux of algo. 1. If both left sub tree and right sub tree cannot lead to path with given sum,Delete the node 2. If any one sub tree can lead to path with sum equal to given sum, do not delete the node */ if(!(leftVal || rightVal) ){ free(node); return false; } if(leftVal || rightVal ){ if(leftVal) node->right = NULL; if(rightVal) node->left = NULL; return true; } return true ; } void inoderTraversal(Node * root){ if(!root) return; inoderTraversal(root->left); printf("%d ", root->value); inoderTraversal(root->right); } Node *createNode(int value){ Node * newNode = (Node *)malloc(sizeof(Node)); newNode->value = value; newNode->right= NULL; newNode->left = NULL; return newNode; } Node *addNode(Node *node, int value){ if(node == NULL){ return createNode(value); } else{ if (node->value > value){ node->left = addNode(node->left, value); } else{ node->right = addNode(node->right, value); } } return node; } /* Driver program for the function written above */ int main(){ Node *root = NULL; //Creating a binary tree root = addNode(root,30); root = addNode(root,20); root = addNode(root,15); root = addNode(root,25); root = addNode(root,40); root = addNode(root,37); root = addNode(root,45); inoderTraversal(root); prunePath(root, 65); printf( "\n"); if( root ){ inoderTraversal(root); } return 0; }

The complexity of this algorithm to prune all nodes which are not on the path with

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