# Lowest Common Ancestor in a Binary Tree in C++

Problem

Given a binary tree, find the lowest common ancestor (LCA) of two given nodes in the tree.

Definition of LCA (according to Wikipedia): The lowest common ancestor is defined between two nodes p and q as the lowest node in T that has both p and q as descendants.

For the given Binary tree, if info[key1] = 15 and info[key2] = 31, it should print:
21
###### Pseudocode
1. Traverse the given binary tree in an preorder fashion.
2. Do if info[root] == key1 or info[root] == key2, return root
3. Traverse root->left and root->right for key1 and key2
4. if key1 and key2, both are found
5.   return (root). Since it is common to the path of both the keys key1 and key2;
6. end

Let the given tree be:

Code Implementation

//
//  main.cpp
//  Lowest Common Ancestor (LCA)
//
//  Created by Himanshu on 31/10/21.
//

#include <iostream>
#include <queue>
using namespace std;

struct node {
int info = 0;
struct node *left, *right;
};
typedef struct node Node;

node* newNode(int data)
{
node* Node = new node();
Node->info = data;
Node->left = NULL;
Node->right = NULL;

return(Node);
}

Node* lca (Node *root, int n1, int n2) {
if (root == NULL) {
return NULL;
}

if (root->info == n1 || root->info == n2) {
return root;
}

Node *leftLCA  = lca(root->left, n1, n2);
Node *rightLCA = lca(root->right, n1, n2);

if (leftLCA && rightLCA) {
return root;
}

return (leftLCA != NULL)? leftLCA : rightLCA;
}

int main() {
Node *root = newNode(1);
root->left = newNode(20);
root->right = newNode(21);
root->right->left = newNode(30);
root->right->right = newNode(31);
root->right->right->left = newNode(40);
root->right->right->left->right = newNode(51);

Node *key = NULL;

cout<<"The lowest common ancestors of 20 and 30 are: "<<endl;
key = lca(root, 20, 30);
cout<<(key->info);
cout<<endl;

cout<<"The lowest common ancestors of 40 and 51 are: "<<endl;
key = lca(root, 40, 51);
cout<<(key->info);
cout<<endl;

return 0;
}



Output

The lowest common ancestors of 20 and 30 are:
1
The lowest common ancestors of 40 and 51 are:
40

Time complexity of the above solution is O(n), where n is the number of nodes in tree. This is because it’s a simple tree traversal.

Here’s a working example: Lowest Common Ancestor