## Question

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:6Explanation: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:42Explanation: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 * 10`

.^{4}] `-1000 <= Node.val <= 1000`

## Python Solution

# Definition for a binary tree node. # class TreeNode: # def __init__(self, val=0, left=None, right=None): # self.val = val # self.left = left # self.right = right max_sum = -2**31 def DFS(root): global max_sum if not root: return -2**31 l = DFS(root.left) r = DFS(root.right) v = root.val max_sum = max(v,v+l,v+r,v+l+r,max_sum) return max(l+v,r+v,v) class Solution: def maxPathSum(self, root: TreeNode) -> int: global max_sum max_sum = -2**31 DFS(root) return max_sum