Maximum Path Sum in a Binary Tree

Given a binary tree, your task is to find the maximum path sum. A path is defined as any sequence of nodes from some starting node to any node in the tree along the parent-child connections. The path must contain at least one node and does not need to go through the root. This problem is fundamental in understanding tree traversals and dynamic programming on trees.

Problem Description

You are given the root of a binary tree. Each node in the tree has an integer value, which can be positive, negative, or zero.

A path in a binary tree is a sequence of nodes where each adjacent pair of nodes in the sequence has an edge connecting them. A node can appear at most once in a path. The path must contain at least one node.

The path sum is the sum of the values of the nodes in a path.

Your goal is to find the maximum path sum among all possible paths in the given binary tree.

Key Requirements:

• The path can start and end at any node in the tree.
• The path does not have to pass through the root.
• The path must contain at least one node.

Edge Cases to Consider:

• An empty tree (though constraints will likely prevent this).
• A tree with only one node.
• Trees with all negative node values.
• Paths that only involve a single node.

Examples

Example 1:

Input: root = [1,2,3]
    1
   / \
  2   3

Output: 6
Explanation: The path 2 -> 1 -> 3 has the maximum sum of 2 + 1 + 3 = 6.

Example 2:

Input: root = [-10,9,20,null,null,15,7]
     -10
     /  \
    9   20
       /  \
      15   7

Output: 42
Explanation: The path 15 -> 20 -> 7 has the maximum sum of 15 + 20 + 7 = 42.

Example 3:

Input: root = [-3]
    -3

Output: -3
Explanation: The only path is the node -3 itself, so the maximum path sum is -3.

Constraints

• The number of nodes in the tree will be between 1 and 3 * 10^4.
• Node values will be between -1000 and 1000.
• The tree is a valid binary tree.

Notes

• This problem can be efficiently solved using a recursive approach, potentially involving a post-order traversal.
• Consider what information each recursive call needs to return to its parent to help calculate the overall maximum path sum.
• Think about how to handle paths that go "up" from a node and then "down" into another subtree.
• You will likely need a global or mutable variable to keep track of the overall maximum path sum found so far.