<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 Info:
    def __init__(self, val):
        self.treeMin = val
        self.treeMax = val
        self.treeSum = val
        self.isBst = True

class Solution:
    def maxSumBST(self, root: Optional[TreeNode]) -> int:
        self.res = -1
        def maxSum(node):
            if not node.left and not node.right:
                self.res = max(self.res, node.val)
                return Info(node.val)

            lInfo = maxSum(node.left) if node.left else None
            rInfo = maxSum(node.right) if node.right else None

            #print(node.val, lInfo.isBst if lInfo else None, rInfo.isBst if rInfo else None, self.res)

            cInfo = Info(node.val)

            if not lInfo and rInfo and rInfo.isBst and node.val < rInfo.treeMin:
                #print('\t 1st else')
                cInfo.treeMin = node.val
                cInfo.treeMax = rInfo.treeMax
                cInfo.treeSum = rInfo.treeSum + node.val
            elif not rInfo and lInfo and lInfo.isBst and node.val > lInfo.treeMax:
                #print('\t 2nd else')
                cInfo.treeMin = lInfo.treeMin
                cInfo.treeMax = node.val
                cInfo.treeSum = node.val + lInfo.treeSum
            elif lInfo and rInfo and lInfo.isBst and rInfo.isBst and node.val > lInfo.treeMax and node.val < rInfo.treeMin:
                #print('\t 3rd else')
                cInfo.treeMin = lInfo.treeMin
                cInfo.treeMax = rInfo.treeMax
                cInfo.treeSum = lInfo.treeSum + rInfo.treeSum + node.val 
            else:
                #print('\t 4th else')
                cInfo.isBst = False
            
            self.res = max(self.res, cInfo.treeSum) if cInfo.isBst else self.res
            #print(cInfo.isBst, self.res)
            return cInfo

        maxSum(root)
        return self.res if self.res > 0 else 0
32900