Given a binary tree, find the maximum path sum.

For this problem, 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 does not need to go through the root.

For example:
Given the below binary tree,

1
2
3
  1
 / \
2   3

Return 6.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
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26
/**
 * Definition for a binary tree node.
 * public class TreeNode {
 *     int val;
 *     TreeNode left;
 *     TreeNode right;
 *     TreeNode(int x) { val = x; }
 * }
 */
public class Solution {
    int maxValue = Integer.MIN_VALUE;
    
    public int maxPathSum(TreeNode root) {
        maxSum(root);
        return maxValue;
    }
    
    public int maxSum(TreeNode root) {
        if (root == null)
            return 0;
        int leftSum = Math.max(0, maxSum(root.left));
        int rightSum = Math.max(0, maxSum(root.right));
        maxValue = Math.max(maxValue, root.val + leftSum + rightSum);
        return root.val + Math.max(leftSum, rightSum);
    }
}