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Pair with a given sum in BST | Set 2
• Difficulty Level : Medium
• Last Updated : 07 Jun, 2021

Given a binary search tree, and an integer X, the task is to check if there exists a pair of distinct nodes in BST with sum equal to X. If yes then print Yes else print No.

Examples:

```Input: X = 5
5
/   \
3     7
/ \   / \
2   4 6   8
Output: Yes
2 + 3 = 5. Thus, the answer is "Yes"

Input: X = 10
1
\
2
\
3
\
4
\
5
Output: No```

Approach: We have already discussed a hash based approach in this article. The space complexity of this is O(N) where N is the number of nodes in BST.

In this article, we will solve the same problem using a space efficient method by reducing the space complexity to O(H) where H is the height of BST. For that, we will use two pointer technique on BST. Thus, we will maintain a forward and a backward iterator that will iterate the BST in the order of in-order and reverse in-order traversal respectively. Following are the steps to solve the problem:

1. Create a forward and backward iterator for BST. Let’s say the value of nodes they are pointing at are v1 and v2.
2. Now at each step,
• If v1 + v2 = X, we found a pair.
• If v1 + v2 < x, we will make forward iterator point to the next element.
• If v1 + v2 > x, we will make backward iterator point to the previous element.
3. If we find no such pair, answer will be “No”.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;` `// Node of the binary tree``struct` `node {``    ``int` `data;``    ``node* left;``    ``node* right;``    ``node(``int` `data)``    ``{``        ``this``->data = data;``        ``left = NULL;``        ``right = NULL;``    ``}``};` `// Function to find a pair with given sum``bool` `existsPair(node* root, ``int` `x)``{``    ``// Iterators for BST``    ``stack it1, it2;` `    ``// Initializing forward iterator``    ``node* c = root;``    ``while` `(c != NULL)``        ``it1.push(c), c = c->left;` `    ``// Initializing backward iterator``    ``c = root;``    ``while` `(c != NULL)``        ``it2.push(c), c = c->right;` `    ``// Two pointer technique``    ``while` `(it1.top() != it2.top()) {` `        ``// Variables to store values at``        ``// it1 and it2``        ``int` `v1 = it1.top()->data, v2 = it2.top()->data;` `        ``// Base case``        ``if` `(v1 + v2 == x)``            ``return` `true``;` `        ``// Moving forward pointer``        ``if` `(v1 + v2 < x) {``            ``c = it1.top()->right;``            ``it1.pop();``            ``while` `(c != NULL)``                ``it1.push(c), c = c->left;``        ``}` `        ``// Moving backward pointer``        ``else` `{``            ``c = it2.top()->left;``            ``it2.pop();``            ``while` `(c != NULL)``                ``it2.push(c), c = c->right;``        ``}``    ``}` `    ``// Case when no pair is found``    ``return` `false``;``}` `// Driver code``int` `main()``{``    ``node* root = ``new` `node(5);``    ``root->left = ``new` `node(3);``    ``root->right = ``new` `node(7);``    ``root->left->left = ``new` `node(2);``    ``root->left->right = ``new` `node(4);``    ``root->right->left = ``new` `node(6);``    ``root->right->right = ``new` `node(8);` `    ``int` `x = 5;` `    ``// Calling required function``    ``if` `(existsPair(root, x))``        ``cout << ``"Yes"``;``    ``else``        ``cout << ``"No"``;` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG``{` `// Node of the binary tree``static` `class` `node``{``    ``int` `data;``    ``node left;``    ``node right;``    ``node(``int` `data)``    ``{``        ``this``.data = data;``        ``left = ``null``;``        ``right = ``null``;``    ``}``};` `// Function to find a pair with given sum``static` `boolean` `existsPair(node root, ``int` `x)``{``    ``// Iterators for BST``    ``Stack it1 = ``new` `Stack(), it2 = ``new` `Stack();` `    ``// Initializing forward iterator``    ``node c = root;``    ``while` `(c != ``null``)``    ``{``        ``it1.push(c);``        ``c = c.left;``    ``}` `    ``// Initializing backward iterator``    ``c = root;``    ``while` `(c != ``null``)``    ``{``        ``it2.push(c);``        ``c = c.right;``    ``}``        ` `    ``// Two pointer technique``    ``while` `(it1.peek() != it2.peek())``    ``{` `        ``// Variables to store values at``        ``// it1 and it2``        ``int` `v1 = it1.peek().data, v2 = it2.peek().data;` `        ``// Base case``        ``if` `(v1 + v2 == x)``            ``return` `true``;` `        ``// Moving forward pointer``        ``if` `(v1 + v2 < x)``        ``{``            ``c = it1.peek().right;``            ``it1.pop();``            ``while` `(c != ``null``)``            ``{``                ``it1.push(c);``                ``c = c.left;``            ``}``        ``}` `        ``// Moving backward pointer``        ``else``        ``{``            ``c = it2.peek().left;``            ``it2.pop();``            ``while` `(c != ``null``)``            ``{``                ``it2.push(c);``                ``c = c.right;``            ``}``        ``}``    ``}``    ` `    ``// Case when no pair is found``    ``return` `false``;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``node root = ``new` `node(``5``);``    ``root.left = ``new` `node(``3``);``    ``root.right = ``new` `node(``7``);``    ``root.left.left = ``new` `node(``2``);``    ``root.left.right = ``new` `node(``4``);``    ``root.right.left = ``new` `node(``6``);``    ``root.right.right = ``new` `node(``8``);` `    ``int` `x = ``5``;` `    ``// Calling required function``    ``if` `(existsPair(root, x))``        ``System.out.print(``"Yes"``);``    ``else``        ``System.out.print(``"No"``);` `}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 implementation of the approach` `# Node of the binary tree``class` `node:``    ` `    ``def` `__init__ (``self``, key):``        ` `        ``self``.data ``=` `key``        ``self``.left ``=` `None``        ``self``.right ``=` `None` `# Function that returns true if a pair``# with given sum exists in the given BSTs``def` `existsPair(root1, x):``    ` `    ``# Stack to store nodes for forward``    ``# and backward iterator``    ``it1, it2 ``=` `[], []` `    ``# Initializing forward iterator``    ``c ``=` `root1``    ``while` `(c !``=` `None``):``        ``it1.append(c)``        ``c ``=` `c.left` `    ``# Initializing backward iterator``    ``c ``=` `root1``    ``while` `(c !``=` `None``):``        ``it2.append(c)``        ``c ``=` `c.right` `    ``# Two pointer technique``    ``while` `(it1[``-``1``] !``=` `it2[``-``1``]):` `        ``# To store the value of the nodes``        ``# current iterators are pointing to``        ``v1 ``=` `it1[``-``1``].data``        ``v2 ``=` `it2[``-``1``].data` `        ``# Base case``        ``if` `(v1 ``+` `v2 ``=``=` `x):``            ``return` `True` `        ``# Moving forward iterator``        ``if` `(v1 ``+` `v2 < x):``            ``c ``=` `it1[``-``1``].right``            ``del` `it1[``-``1``]``            ` `            ``while` `(c !``=` `None``):``                ``it1.append(c)``                ``c ``=` `c.left` `        ``# Moving backward iterator``        ``else``:``            ``c ``=` `it2[``-``1``].left``            ``del` `it2[``-``1``]``            ` `            ``while` `(c !``=` `None``):``                ``it2.append(c)``                ``c ``=` `c.right` `    ``# If no such pair found``    ``return` `False` `# Driver code``if` `__name__ ``=``=` `'__main__'``:` `    ``root2 ``=` `node(``5``)``    ``root2.left ``=` `node(``3``)``    ``root2.right ``=` `node(``7``)``    ``root2.left.left ``=` `node(``2``)``    ``root2.left.right ``=` `node(``4``)``    ``root2.right.left ``=` `node(``6``)``    ``root2.right.right ``=` `node(``8``)` `    ``x ``=` `5``    ` `    ``# Calling required function``    ``if` `(existsPair(root2, x)):``        ``print``(``"Yes"``)``    ``else``:``        ``print``(``"No"``)` `# This code is contributed by mohit kumar 29`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{` `// Node of the binary tree``public` `class` `node``{``    ``public` `int` `data;``    ``public` `node left;``    ``public` `node right;``    ``public` `node(``int` `data)``    ``{``        ``this``.data = data;``        ``left = ``null``;``        ``right = ``null``;``    ``}``};` `// Function to find a pair with given sum``static` `bool` `existsPair(node root, ``int` `x)``{``    ``// Iterators for BST``    ``Stack it1 = ``new` `Stack(),``                 ``it2 = ``new` `Stack();` `    ``// Initializing forward iterator``    ``node c = root;``    ``while` `(c != ``null``)``    ``{``        ``it1.Push(c);``        ``c = c.left;``    ``}` `    ``// Initializing backward iterator``    ``c = root;``    ``while` `(c != ``null``)``    ``{``        ``it2.Push(c);``        ``c = c.right;``    ``}``        ` `    ``// Two pointer technique``    ``while` `(it1.Peek() != it2.Peek())``    ``{` `        ``// Variables to store values at``        ``// it1 and it2``        ``int` `v1 = it1.Peek().data,``            ``v2 = it2.Peek().data;` `        ``// Base case``        ``if` `(v1 + v2 == x)``            ``return` `true``;` `        ``// Moving forward pointer``        ``if` `(v1 + v2 < x)``        ``{``            ``c = it1.Peek().right;``            ``it1.Pop();``            ``while` `(c != ``null``)``            ``{``                ``it1.Push(c);``                ``c = c.left;``            ``}``        ``}` `        ``// Moving backward pointer``        ``else``        ``{``            ``c = it2.Peek().left;``            ``it2.Pop();``            ``while` `(c != ``null``)``            ``{``                ``it2.Push(c);``                ``c = c.right;``            ``}``        ``}``    ``}``    ` `    ``// Case when no pair is found``    ``return` `false``;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``node root = ``new` `node(5);``    ``root.left = ``new` `node(3);``    ``root.right = ``new` `node(7);``    ``root.left.left = ``new` `node(2);``    ``root.left.right = ``new` `node(4);``    ``root.right.left = ``new` `node(6);``    ``root.right.right = ``new` `node(8);` `    ``int` `x = 5;` `    ``// Calling required function``    ``if` `(existsPair(root, x))``        ``Console.Write(``"Yes"``);``    ``else``        ``Console.Write(``"No"``);``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``
Output:
`Yes`

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