## # 124. Binary Tree Maximum Path Sum

Given a non-empty 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 must contain at least one node and does not need to go through the root.

Example 1:

``````Input: [1,2,3]

1
/ \
2   3

Output: 6
``````

Example 2:

``````Input: [-10,9,20,null,null,15,7]

-10
/ \
9  20
/  \
15   7

Output: 42
``````

### # Solution

Approach 1: once we know the maximum sum of each node (node itself and a simple path down its subtree, no turns) in the tree, the max path sum can be found recursively from 3 cases: when the result passes the root node, when the result is in the right subtree only, when the result is in the left subtree only. Time: `O(N)`.

### # Code (Python)

Approach 1:

``````class Solution:
def maxPathSum(self, root):
"""
:type root: TreeNode
:rtype: int
"""
return self._max_path_sum(root, {root: root.val, None: 0})

def _max_path_sum(self, root, cache):
if not root:
return 0
# when result passes the root node
result = root.val + max(0, self._max_sum_from_root(root.left, cache)) + max(0, self._max_sum_from_root(root.right, cache))
# when result is in the left subtree
if root.left:
result = max(result, self._max_path_sum(root.left, cache))
# when result is in the right subtree
if root.right:
result = max(result, self._max_path_sum(root.right, cache))
return result

def _max_sum_from_root(self, root, cache): # the cache stores _max_sum_from_root() for each node
# calculate the maximum sum of each node (node itself and a simple path down its subtree, no turns)
if not root:
return 0
if root.left not in cache:
cache[root.left] = self._max_sum_from_root(root.left, cache)
if root.right not in cache:
cache[root.right] = self._max_sum_from_root(root.right, cache)
# 0 needs to be compared also because values can be negative
return max(cache[root.left], cache[root.right], 0) + root.val
``````

### # Code (C++)

Approach 1:

``````/**
* Definition for a binary tree node.
* struct TreeNode {
*     int val;
*     TreeNode *left;
*     TreeNode *right;
*     TreeNode(int x) : val(x), left(NULL), right(NULL) {}
* };
*/
class Solution {
private:
int maxPath;
int doMaxPathSum(TreeNode* root) {
if (root == NULL)
return 0;
int left = doMaxPathSum(root->left);
int right = doMaxPathSum(root->right);
int res = root->val;
if (left > 0 || right > 0)
res += std::max(left, right);
maxPath = std::max(maxPath, res);
if (left > 0 && right > 0)
maxPath = std::max(maxPath, left + right + root->val);
return res;
}
public:
int maxPathSum(TreeNode* root) {
maxPath = INT_MIN;
doMaxPathSum(root);
return maxPath;
}
};
``````