<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:
        return max(0, traverse(root)[2])

def traverse(root):
    if root.left is None and root.right is None:
        return True, root.val, root.val, root.val, root.val
    
    if root.left is None:
        bst, bst_sum, max_sum, min_val, max_val = traverse(root.right)
        if bst:
            if root.val < min_val:
                return True, root.val + bst_sum, max(root.val + bst_sum, max_sum), root.val, max_val
            return False, 0, max_sum, 0, 0
        return False, 0, max_sum, 0, 0

    if root.right is None:
        bst, bst_sum, max_sum, min_val, max_val = traverse(root.left)
        if bst:
            if root.val > max_val:
                return True, root.val + bst_sum, max(root.val + bst_sum, max_sum), min_val, root.val
            return False, 0, max_sum, 0, 0
        return False, 0, max_sum, 0, 0
    
    bst1, bst_sum1, max_sum1, min_val1, max_val1 = traverse(root.left)
    bst2, bst_sum2, max_sum2, min_val2, max_val2 = traverse(root.right)

    if bst1 and bst2 and (root.val > max_val1) and (root.val < min_val2):
        return True, bst_sum1 + bst_sum2 + root.val, max(bst_sum1 + bst_sum2 + root.val, max_sum1, max_sum2), min_val1, max_val2
    
    return False, 0, max(max_sum1, max_sum2), 0, 0
37300