<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 T:
  def __init__(self, isBST: bool = False,
               max: Optional[int] = None,
               min: Optional[int] = None,
               sum: Optional[int] = None):
    self.isBST = isBST
    self.max = max
    self.min = min
    self.sum = sum


class Solution:
  def maxSumBST(self, root: Optional[TreeNode]) -> int:
    self.ans = 0

    def traverse(root: Optional[TreeNode]) -> T:
        if not root:
            return T(True, -math.inf, math.inf, 0)

        left: T = traverse(root.left)
        right: T = traverse(root.right)

        # subtrees are valid BST
        if not left.isBST or not right.isBST:
            return T()

        # check if curr is valid BST
        if root.val <= left.max or root.val >= right.min:
            return T()

        summ = root.val + left.sum + right.sum
        self.ans = max(self.ans, summ)

        # taking the max due to base case
        return T(True, max(root.val, right.max), min(root.val, left.min), summ)

    traverse(root)
    return self.ans
305