Binary Tree Maximum Path Sum

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:** **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:** **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 `[1, 3 * 104]`. - `-1000 <= Node.val <= 1000`