124\. Binary Tree Maximum Path Sum

Hard

A **path** in a binary tree is a sequence of nodes where each pair of adjacent nodes in the sequence has an edge connecting them. A node can only appear in the sequence **at most once**. Note that the path does not need to pass through the root.

The **path sum** of a path is the sum of the node's values in the path.

Given the `root` of a binary tree, return _the maximum **path sum** of any **non-empty** path_.

**Example 1:**

![](https://assets.leetcode.com/uploads/2020/10/13/exx1.jpg)

**Input:** root = [1,2,3]

**Output:** 6

**Explanation:** The optimal path is 2 -> 1 -> 3 with a path sum of 2 + 1 + 3 = 6. 

**Example 2:**

![](https://assets.leetcode.com/uploads/2020/10/13/exx2.jpg)

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

**Output:** 42

**Explanation:** The optimal path is 15 -> 20 -> 7 with a path sum of 15 + 20 + 7 = 42. 

**Constraints:**

*   The number of nodes in the tree is in the range <code>[1, 3 * 10<sup>4</sup>]</code>.
*   `-1000 <= Node.val <= 1000`