Lowest Common Ancestor In A Binary Tree

Lowest Common Ancestor in a Binary Tree: A complete walkthrough

The lowest common ancestor (LCA) in a binary tree is a fundamental concept in computer science and data structures, frequently encountered in algorithm design, coding interviews, and real-world applications like file system navigation or network routing. Understanding how to compute the LCA efficiently is crucial for solving complex tree-based problems. This article explores the definition, approaches, and practical implementations of finding the LCA in a binary tree, providing a thorough foundation for both beginners and experienced developers.

Understanding the Problem

What is the Lowest Common Ancestor?

In a binary tree, the lowest common ancestor (LCA) of two nodes u and v is the deepest node that has both u and v as descendants (where a node can be a descendant of itself). Take this: consider the following binary tree:

        1
       / \
      2   3
     / \
    4   5

• The LCA of nodes 4 and 5 is 2.
• The LCA of nodes 4 and 3 is 1.
• The LCA of nodes 2 and 4 is 2 (since a node is considered a descendant of itself).

Key Observations

1. Self-Descendant Rule: A node can be its own ancestor.
2. Existence Check: The problem assumes both nodes exist in the tree. If not, additional validation is required.
3. Binary Tree Structure: The tree is not necessarily a binary search tree (BST), so node values do not follow ordering constraints.

Approaches to Find the Lowest Common Ancestor

1. Brute-Force Approach (Path Comparison)

The simplest way to find the LCA is by finding the paths from the root to both nodes and then comparing the paths to identify their last common node.

Steps:

1. Find Path to Node 1: Traverse the tree from root to node u, storing the path in a list.
2. Find Path to Node 2: Traverse the tree from root to node v, storing the path in a list.
3. Compare Paths: Iterate through both paths simultaneously until they diverge. The last common node is the LCA.

Time and Space Complexity:

• Time: O(n), where n is the number of nodes (due to two traversals).
• Space: O(h), where h is the height of the tree (for storing paths).

Pseudocode:

def find_lca(root, u, v):
    path_u = find_path(root, u)
    path_v = find_path(root