<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:
        valid = {}
        SUM = {} 
        self.processTree(root, valid, SUM)
        res = []
        for k,v in SUM.items():
            if valid[k]:
                res.append(v)
        ans = max(res)
        if ans>0:
            return ans
        return 0


    def processTree(self, root, valid, SUM):
        if root.right!=None:
            maxValRt, minValRt, isRtBSt, sumRt = self.processTree(root.right, valid, SUM)
        else:
            maxValRt, minValRt, isRtBSt, sumRt = float('inf'), float('inf'),True,0
        if root.left != None:
            maxValLt, minValLt, isLtBst, sumLt = self.processTree(root.left, valid, SUM)
        else:
            maxValLt, minValLt, isLtBst, sumLt = float('-inf'), float('-inf'),True,0
        print(root.val, maxValRt, minValRt, maxValLt, minValLt, isRtBSt, isLtBst)
        valid[root] = root.val<minValRt and root.val>maxValLt and isRtBSt and isLtBst
        SUM[root] = sumRt+sumLt+root.val
        if max(maxValRt, maxValLt, root.val) == float('inf'):
            if min(minValLt, minValRt, root.val) == float('-inf'):
                return root.val, root.val, valid[root], SUM[root]
            else:
                return root.val, min(minValLt, minValRt, root.val), valid[root], SUM[root]
        if min(minValLt, minValRt, root.val) == float('-inf'):
            if max(maxValRt, maxValLt, root.val) == float('inf'):
                return root.val, root.val, valid[root], SUM[root]
            else:
                return max(root.val, maxValRt, maxValLt),root.val,  valid[root], SUM[root]
        return max(root.val, maxValRt, maxValLt), min(root.val, minValLt, minValRt),  valid[root], SUM[root]
        
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