<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:
        maxsum = [0]
        def postorder(node):
            if node.left is None and node.right is None:
                maxsum[0] = max(maxsum[0],node.val)
                return (node.val,node.val,node.val)
            if node.left is None:            
                t = postorder(node.right)
                if(node.val < t[1]):
                    maxsum[0] = max(maxsum[0],node.val + t[0])
                    return (t[0]+node.val,node.val,t[2])
                return (t[0],float('-inf'),float('inf'))
            if node.right is None:            
                t = postorder(node.left)
                if(node.val > t[2]):
                    maxsum[0] = max(maxsum[0],node.val + t[0])
                    return (t[0]+node.val,t[1],node.val)
                return (t[0],float('-inf'),float('inf'))
            tl = postorder(node.left)
            tr = postorder(node.right)
            if node.val == 1:
                print(tl)
                print(tr)
            if(node.val > tl[2] and node.val < tr[1]):
                maxsum[0] = max(maxsum[0],node.val + tl[0] + tr[0])
                return (tl[0]+tr[0]+node.val,tl[1],tr[2])
            else:
                return (0,float('-inf'),float('inf'))
        t = postorder(root)
        return maxsum[0]
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