<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:
        self.max_bst_sum = 0
        if root is not None:
            self._dfs_max_bst_sum(root)
        return max(self.max_bst_sum, 0)  # Required to set 0 as floor.

    def _dfs_max_bst_sum(self, current_node: TreeNode):
        if not current_node.left and not current_node.right:  # Leaf node.
            if current_node.val > self.max_bst_sum:
                self.max_bst_sum = current_node.val
            return current_node.val, current_node.val, current_node.val
        
        if current_node.left:
            left_bst_sum, left_bst_min, left_bst_max = self._dfs_max_bst_sum(current_node.left)
        
        else:
            left_bst_sum, left_bst_min, left_bst_max = 0, -float("inf"), -float("inf")
        
        if current_node.right:
            right_bst_sum, right_bst_min, right_bst_max = self._dfs_max_bst_sum(current_node.right)
        
        else:
            right_bst_sum, right_bst_min, right_bst_max = 0, float("inf"), float("inf")
        
        if None in {left_bst_sum, right_bst_sum}:  # Foul: current node's BST nullified.
            current_bst_sum = None  # To notify parent node during return.

        else:
            # Valid BST: max left child < non-leaf node < min right child.
            if left_bst_max < current_node.val < right_bst_min:  # Current node's BST stands.
                current_bst_sum = left_bst_sum + right_bst_sum + current_node.val
                if current_bst_sum > self.max_bst_sum:  # Current BST may set new record.
                    self.max_bst_sum = current_bst_sum
        
            else:  # Foul: current node's BST nullified.
                current_bst_sum = None  # To notify parent node during return.

        # Find current BST's min & max (don't consider infinities).
        current_bst_min, current_bst_max = current_node.val, current_node.val
        for value in {left_bst_min, left_bst_max, right_bst_min, right_bst_max}:
            if value in {-float("inf"), float("inf")}:
                continue
            
            if value < current_bst_min:
                current_bst_min = value
            if value > current_bst_max:
                current_bst_max = value
        
        return current_bst_sum, current_bst_min, current_bst_max

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