<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 NodeValue:
    def __init__(self, min_val, max_val, sum):
        self.min_val = min_val
        self.max_val = max_val
        self.sum = sum

class Solution:
    def maxSumBST(self, root: Optional[TreeNode]) -> int:
        # def isBST(node):
        #     if not node:
        #         return True

        #     def fn(root, min_val, max_val):
        #         if not root:
        #             return True
                
        #         if min_val >= root.val or root.val >= max_val:
        #             return False
                
        #         l = fn(root.left, min_val, root.val)
        #         r = fn(root.right, root.val, max_val)

        #         return l and r

        #     return fn(node, -math.inf, math.inf)
        
        # def total_sum_BST(node, total_sum):
        #     if not node:
        #         return 0
            
        #     total_sum[0] += node.val
        
        #     total_sum_BST(node.left, total_sum)
        #     total_sum_BST(node.right, total_sum)

        #     return total_sum[0]

        # def max_sum(node, res):
        #     if not node:
        #         return

        #     if isBST(node):
        #         temp = [0]
        #         res[0] = max(res[0], total_sum_BST(node, temp))
            
        #     max_sum(node.left, res)
        #     max_sum(node.right, res)

        
        # res = [0]

        # max_sum(root, res)
        # return res[0]


        ##### OPTIMAL APPROACH
        
        def total_sum_BST(root):
            if not root:
                return NodeValue(math.inf, -math.inf, 0)
            
        
            l = total_sum_BST(root.left)
            r = total_sum_BST(root.right)

            if l.max_val < root.val < r.min_val:
                curr_sum = l.sum + r.sum + root.val
                self.res = max(self.res, curr_sum)
                return NodeValue(min(l.min_val, root.val), max(r.max_val, root.val), curr_sum)
            
            
            return NodeValue(-math.inf, math.inf, 0)

        
        self.res = 0
        total_sum_BST(root)

        return self.res


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