<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:
        if not root:
            return 0
        
        largest = [0]
        self.largest_sum_bst(root)
        self.dfs(root, largest)
        return largest[-1]

    def largest_sum_bst(self, node):
        if not node:
            new_node = TreeNode(None, None, None)
            new_node.is_bst = True
            new_node.sum = 0
            new_node.min = +inf
            new_node.max = -inf
            return new_node
        
        if not node.left and not node.right:
            node.is_bst = True
            node.sum = node.val
            node.min = node.val
            node.max = node.val
            return node
        else:
            node.sum = node.val
            node.min = node.val
            node.max = node.val
            node.is_bst = True
            left_bst = self.largest_sum_bst(node.left)
            right_bst = self.largest_sum_bst(node.right)
            if left_bst.is_bst and right_bst.is_bst and node.val > left_bst.max and node.val < right_bst.min:
                node.sum += left_bst.sum + right_bst.sum
                node.min = min(node.min, left_bst.min)
                node.max = max(node.max, right_bst.max)
                return node
            else:
                node.is_bst = False
                node.min = -inf
                node.max = +inf
                node.sum = max(left_bst.sum, right_bst.sum)
                return node
    
    def dfs(self, node, largest):
        if not node:
            return
        if node.is_bst:
            largest[-1] = max(largest[-1], node.sum)
        if node.left:
            self.dfs(node.left, largest)
        
        if node.right:
            self.dfs(node.right, largest)
37600