<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 Solution:
    class Box:
        def __init__(self):
            self.BST = True
            self.sum = 0
            self.min = float('inf')
            self.max = float('-inf')

    def find(self, root, totalSum):
        if not root:
            return self.Box()

        leftBox = self.find(root.left, totalSum)
        rightBox = self.find(root.right, totalSum)

        if leftBox.BST and rightBox.BST and leftBox.max < root.val < rightBox.min:
            currentBox = self.Box()
            currentBox.sum = leftBox.sum + rightBox.sum + root.val
            currentBox.min = min(root.val, leftBox.min)
            currentBox.max = max(root.val, rightBox.max)

            totalSum[0] = max(totalSum[0], currentBox.sum)

            return currentBox
        else:
            invalidBox = self.Box()
            invalidBox.BST = False
            return invalidBox

    def maxSumBST(self, root: TreeNode) -> int:
        totalSum = [0]
        self.find(root, totalSum)
        return totalSum[0]
334