<p>Given a <strong>binary tree</strong> <code>root</code>, return <em>the maximum sum of all keys of <strong>any</strong> sub-tree which is also a Binary Search Tree (BST)</em>.</p>

<p>Assume a BST is defined as follows:</p>

<ul>
	<li>The left subtree of a node contains only nodes with keys <strong>less than</strong> the node&#39;s key.</li>
	<li>The right subtree of a node contains only nodes with keys <strong>greater than</strong> the node&#39;s key.</li>
	<li>Both the left and right subtrees must also be binary search trees.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/01/30/sample_1_1709.png" style="width: 320px; height: 250px;" /></p>

<pre>
<strong>Input:</strong> root = [1,4,3,2,4,2,5,null,null,null,null,null,null,4,6]
<strong>Output:</strong> 20
<strong>Explanation:</strong> Maximum sum in a valid Binary search tree is obtained in root node with key equal to 3.
</pre>

<p><strong class="example">Example 2:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/01/30/sample_2_1709.png" style="width: 134px; height: 180px;" /></p>

<pre>
<strong>Input:</strong> root = [4,3,null,1,2]
<strong>Output:</strong> 2
<strong>Explanation:</strong> Maximum sum in a valid Binary search tree is obtained in a single root node with key equal to 2.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> root = [-4,-2,-5]
<strong>Output:</strong> 0
<strong>Explanation:</strong> All values are negatives. Return an empty BST.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li>The number of nodes in the tree is in the range <code>[1, 4 * 10<sup>4</sup>]</code>.</li>
	<li><code>-4 * 10<sup>4</sup> &lt;= Node.val &lt;= 4 * 10<sup>4</sup></code></li>
</ul>

# 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
class Solution:
    def maxSumBST(self, root: Optional[TreeNode]) -> int:
        maxSum = -4 * 10**4
                 # subtree: is BST, sum, minVal, maxVal
        def dfs(node) -> tuple[bool, int, int, int]:
            nonlocal maxSum
            if node.val >= 0: allNeg = False
            if not node.left and not node.right:
                maxSum = max(maxSum, node.val)
                return True, node.val, node.val, node.val

            if node.left and node.right:
                isBstL, sumL, minL, maxL = dfs(node.left)
                isBstR, sumR, minR, maxR = dfs(node.right)
                isBst = isBstL and isBstR and maxL < node.val < minR
                if isBst:
                    currSum = sumL + node.val + sumR
                    maxSum = max(maxSum, currSum)
                    return True, currSum, minL, maxR

            elif node.left:
                isBstC, sumC, minC, maxC = dfs(node.left)
                isBst = isBstC and maxC < node.val
                if isBst:
                    currSum = sumC + node.val
                    maxSum = max(maxSum, currSum)
                    return True, currSum, minC, node.val
            
            elif node.right:
                isBstC, sumC, minC, maxC = dfs(node.right)
                isBst = isBstC and node.val < minC
                if isBst:
                    currSum = sumC + node.val
                    maxSum = max(maxSum, currSum)
                    return True, currSum, node.val, maxC

            return False, 0, 0, 0

        dfs(root)
        # It is unclear why tests expect the max sum to never be under 0
        # Example 3 says "All values are negatives. Return an empty BST.".
        # This is not obvious from the problem statement though.
        return max(maxSum, 0)

            
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