<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 TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

class Solution:
    class MyNode:
        def __init__(self, maxNode, minNode, maxSum):
            self.maxNode = maxNode
            self.minNode = minNode
            self.maxSum = maxSum

    def __init__(self):
        self.MAX_SUM = float('-inf')

    def getMaxSum(self, root):
        if not root:
            return self.MyNode(float('-inf'), float('inf'), 0)

        myLeftNode = self.getMaxSum(root.left)
        myRightNode = self.getMaxSum(root.right)

        if myLeftNode.maxNode < root.val < myRightNode.minNode:
            self.MAX_SUM = max(self.MAX_SUM, myLeftNode.maxSum + myRightNode.maxSum + root.val)

            return self.MyNode(
                max(root.val, myRightNode.maxNode),
                min(root.val, myLeftNode.minNode),
                myLeftNode.maxSum + myRightNode.maxSum + root.val
            )

        return self.MyNode(float('inf'), float('-inf'), 0)

    def maxSumBST(self, root):
        node = self.getMaxSum(root)
        return 0 if self.MAX_SUM < 0 else self.MAX_SUM


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