###### tags: `leetcode` `hard` `Dynamic Programming` `Tree` `Depth-First Search` `Binary Tree`
# [124. Binary Tree Maximum Path Sum](https://leetcode.com/problems/binary-tree-maximum-path-sum/description/)

## Description

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*.

## Examples
### 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 $[1, 3 \times 10^4]$.
- $-1000 \leq Node.val \leq 1000$

## Code

/**
 * Definition for a binary tree node.
 * struct TreeNode {
 *     int val;
 *     struct TreeNode *left;
 *     struct TreeNode *right;
 * };
 */
typedef struct TreeNode TNODE;
#define CASE_MIN -1001
#define MAX(x, y) ((x) > (y) ? (x) : (y))

// Find the depth from root to leaf
int DEPTH_MAX_SEARCH(TNODE* root, int* crossMax) {
    // If node is null, return CASE_MIN
    if (!root)
        return CASE_MIN;
    int left = DEPTH_MAX_SEARCH(root->left, crossMax);
    int right = DEPTH_MAX_SEARCH(root->right, crossMax);
    // No-child leaf return its value
    if (left == CASE_MIN && right == CASE_MIN)
        return root->val;
    int maxChild = MAX(left, right);
    // Avoid child having CASE_MIN, using maxChild as filter
    *crossMax = MAX(*crossMax, maxChild);
    *crossMax = MAX(*crossMax, root->val);
    *crossMax = MAX(*crossMax, (left + right + root->val));
    return MAX((maxChild + root->val), root->val);
}

int maxPathSum(struct TreeNode* root) {
    int crossMax = CASE_MIN;
    int depthMax = DEPTH_MAX_SEARCH(root, &crossMax);
    return MAX(crossMax, depthMax);
}

## Complexity

|Space |Time |
|- |- |
|$O(1)$|$O(N)$|

## Result

- Runtime : 18 ms, 95.54%
- Memory : 13.6 MB, 50.89%