<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:
        # O( V + E) ~ O(N)
        max_sum = float('-inf')
        
        def dfs(root):
            nonlocal max_sum
            if not root:
                return (0, float('inf'), float('-inf'), True)
            
            ls, left_min, left_max, left_balanced = dfs(root.left)
            rs, right_min, right_max, right_balanced = dfs(root.right)
            
            if left_balanced and right_balanced and left_max < root.val < right_min:
                
                tree_sum = ls + rs + root.val
                max_sum = max(max_sum, tree_sum)
                lower_bound =  min(left_min, root.val)
                upper_bound = max(right_max, root.val)
                
                return (tree_sum, lower_bound, upper_bound, True)
            
            return (0, float('inf'), float('-inf'), False)
        
        
        '''
        def bst(root, lower=float('inf'), upper=float('-inf')):
            if not root:
                return True
            left = bst(root.left, lower, root.val)
            right = bst(root.right, root.val, upper)
            return left and right and lower < root.val < upper
        '''
        
        max_sum = 0
        dfs(root)
        return max_sum
    
    
    
37700