<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>

class Solution:
    def maxSumBST(self, root: Optional[TreeNode]) -> int:
        self.maxsum = None
        self.dfs(root)
        return max(self.maxsum, 0)
    
    def dfs(self, root: TreeNode) -> None:
        if root.left:
            self.dfs(root.left)
        if root.right:
            self.dfs(root.right)

        if not root.left and not root.right:
            # min, max, sum
            root.val = [root.val, root.val, root.val]
        else:
            if (root.left and not root.left.val) or root.left and root.left.val and root.left.val[1] >= root.val:
                root.val = None
            elif (root.right and not root.right.val) or root.right and root.right.val and root.right.val[0] <= root.val:
                root.val = None
            else:
                val = [root.val, root.val, root.val]
                if root.left and root.left.val:
                    val[0] = root.left.val[0]
                    val[2] += root.left.val[2]
                if root.right and root.right.val:
                    val[1] = root.right.val[1]
                    val[2] += root.right.val[2]
                root.val = val
        
        root.left = None
        root.right = None
        
        if root.val:
            if not self.maxsum:
                self.maxsum = root.val[2]
            else:
                self.maxsum = max(self.maxsum, root.val[2])

192