<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:
        final_res = 0
        def helper(node):
            nonlocal final_res
            # returns a (bool, min, max, node_sum)
            if node.left == None and node.right==None:
                final_res = max(final_res, node.val)
                return (True, node.val, node.val, node.val)
            if node.left == None or node.right==None:
                res = None
                if node.left == None:
                    res = helper(node.right)
                    if res[0] and node.val < res[1]:
                        res = (True, node.val, res[2], node.val+res[3])
                        final_res = max(final_res, res[3])
                    else:
                        res = (False, res[1], res[2], res[3])
                else: 
                    res = helper(node.left)
                    if res[0] and node.val > res[2]:
                        res = (True, res[1], node.val, node.val+res[3])
                        final_res = max(final_res, res[3])
                    else:
                        res = (False, res[1], res[2], res[3])
                print(res)
                return res
            l, r = helper(node.left), helper(node.right)
            if l[3] >= r[3]:
                res = (False, l[1], l[2], l[3])
            else:
                res = (False, r[1], r[2], r[3])
            if l[0] and r[0] and node.val < r[1] and node.val > l[2]:
                res = (True, l[1], r[2], l[3]+r[3]+node.val)
                final_res = max(final_res, res[3])
            print(res)
            return res
        helper(root)
        return final_res

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