# Lowest Common Ancestor of a Binary Search Tree

Published: Oct 29, 2022

Medium Depth-First Search Binary Search Tree

## Problem Description

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

According to the definition of LCA on 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 (where we allow a node to be a descendant of itself).”

Constraints:

• The number of nodes in the tree is in the range `[2, 10**5]`.
• `-10**9 <= Node.val <= 10**9`
• All Node.val are unique.
• `p != q`
• `p` and `q` will exist in the BST.

https://leetcode.com/problems/lowest-common-ancestor-of-a-binary-search-tree/

## Examples

``````Example 1
Input: root = [6,2,8,0,4,7,9,null,null,3,5], p = 2, q = 8
Output: 6
Explanation: The LCA of nodes 2 and 8 is 6.
6
/   \
2      8
/ \    / \
0   4  7   9
/ \
3   5
``````
``````Example 2
Input: root = [6,2,8,0,4,7,9,null,null,3,5], p = 2, q = 4
Output: 2
Explanation: The LCA of nodes 2 and 4 is 2, since a node can be a descendant of itself according to the LCA definition.
6
/   \
2      8
/ \    / \
0   4  7   9
/ \
3   5
``````
``````Example 3
Input: root = [2,1], p = 2, q = 1
Output: 2
``````

## How to Solve

This is a popular LCA problem of the binary search tree (BST). The binary tree’s LCA solution is Lowest Common Ancestor of a Binary Tree. Either of DFS or BFS works to find the solution. Compare to the binary tree, the BST can find the answer quickly in general. That’s because BST can check the node value and decide which subtree to go deeper. The solution here took the DFS approach. If both p and q values are smaller than root value, it should check left subtree. Otherwise, go right subtree. If the root value lies between p and q node values, it is a lca.

## Solution

``````/**
* Definition for a binary tree node.
* struct TreeNode {
*     int val;
*     TreeNode *left;
*     TreeNode *right;
*     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
* };
*/

class LowestCommonAncestorOfABinarySearchTree {
public:
TreeNode* lowestCommonAncestor(TreeNode* root, TreeNode* p, TreeNode* q) {
if (!root) {
return root;
}
if (p->val < root->val && q->val < root->val) {
return lowestCommonAncestor(root->left, p, q);
} else if (root->val < p->val && root->val < q->val){
return lowestCommonAncestor(root->right, p, q);
} else {
return root;
}
}
};
``````
``````
``````
``````
``````
``````# Definition for a binary tree node.
# class TreeNode:
#     def __init__(self, x):
#         self.val = x
#         self.left = None
#         self.right = None

class LowestCommonAncestorOfABinarySearchTree:
def lowestCommonAncestor(self, root: 'TreeNode', p: 'TreeNode', q: 'TreeNode') -> 'TreeNode':
if not root:
return root
if p.val < root.val and q.val < root.val:
return self.lowestCommonAncestor(root.left, p, q)
elif root.val < p.val and root.val < q.val:
return self.lowestCommonAncestor(root.right, p, q)
else:
return root
``````
``````
``````

## Complexities

• Time: `O(n)` – if the BST is well balanced, this can be log(n)
• Space: `O(n)`