Lowest Common Ancestor of a Binary Tree III

This problem challenges you to find the lowest common ancestor (LCA) of two given nodes within a binary tree. Unlike finding the LCA of two nodes in a standard binary tree, here the tree is represented as a linked list of nodes, where each node can have up to three child pointers (n1, n2, n3). Successfully solving this problem demonstrates proficiency in tree traversal and understanding of LCA concepts in a less conventional tree structure.

Problem Description

You are given a binary tree represented as a linked list of nodes. Each node in the tree has a val field representing its value and three child pointers: n1, n2, and n3. You are also given two nodes, p and q, which are guaranteed to be present in the tree. The task is to find the lowest common ancestor (LCA) of nodes p and q.

The LCA of two nodes p and q is defined as the lowest node in the tree such that both p and q are descendants of this node (where "descendant" means the node is on the path from the LCA to p or q).

Requirements:

• Write a function that takes the root node of the tree and the two nodes p and q as input.
• The function should return the LCA node of p and q.
• The function should handle cases where p and q are the same node.
• The function should handle cases where one of the nodes is an ancestor of the other.
• The function should efficiently traverse the tree to find the LCA.

Expected Behavior:

The function should return the node that is the lowest common ancestor of p and q. If p and q are the same node, that node is the LCA. If either p or q is not in the tree, the behavior is undefined (though the problem statement guarantees they are present).

Examples

Example 1:

Input: root = [3,5,1,6,2,0,8,None,None,7,4], p = 5, q = 1
Output: 3
Explanation: The LCA of nodes 5 and 1 is 3.

Example 2:

Input: root = [3,5,1,6,2,0,8,None,None,7,4], p = 5, q = 5
Output: 5
Explanation: The LCA of nodes 5 and 5 is 5.

Example 3:

Input: root = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], p = 8, q = 15
Output: 1
Explanation: The LCA of nodes 8 and 15 is 1.

Constraints

• The number of nodes in the tree is between 1 and 10⁵.
• The values of the nodes are unique.
• p and q are guaranteed to be nodes in the tree.
• The tree is a valid binary tree with up to three children per node.

Notes

Consider using a depth-first search (DFS) approach to traverse the tree. You can track the path from the root to each node and then compare the paths to find the LCA. Think about how to efficiently determine if a node is an ancestor of another node. The three child pointers introduce a slight complexity compared to a standard binary tree, so be sure to handle all possible paths correctly. A recursive solution is often a good approach for tree traversal problems.