Added max_sum_bst.py (#11832)

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* Update maximum_sum_bst.py

* def max_sum_bst(root: TreeNode | None) -> int:

* def solver(node: TreeNode | None) -> tuple[bool, int, int, int]:

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>

data_structures/binary_tree/maximum_sum_bst.py

@@ -0,0 +1,78 @@
+from __future__ import annotations
+
+import sys
+from dataclasses import dataclass
+
+INT_MIN = -sys.maxsize + 1
+INT_MAX = sys.maxsize - 1
+
+
+@dataclass
+class TreeNode:
+    val: int = 0
+    left: TreeNode | None = None
+    right: TreeNode | None = None
+
+
+def max_sum_bst(root: TreeNode | None) -> int:
+    """
+    The solution traverses a binary tree to find the maximum sum of
+    keys in any subtree that is a Binary Search Tree (BST). It uses
+    recursion to validate BST properties and calculates sums, returning
+    the highest sum found among all valid BST subtrees.
+
+    >>> t1 = TreeNode(4)
+    >>> t1.left = TreeNode(3)
+    >>> t1.left.left = TreeNode(1)
+    >>> t1.left.right = TreeNode(2)
+    >>> print(max_sum_bst(t1))
+    2
+    >>> t2 = TreeNode(-4)
+    >>> t2.left = TreeNode(-2)
+    >>> t2.right = TreeNode(-5)
+    >>> print(max_sum_bst(t2))
+    0
+    >>> t3 = TreeNode(1)
+    >>> t3.left = TreeNode(4)
+    >>> t3.left.left = TreeNode(2)
+    >>> t3.left.right = TreeNode(4)
+    >>> t3.right = TreeNode(3)
+    >>> t3.right.left = TreeNode(2)
+    >>> t3.right.right = TreeNode(5)
+    >>> t3.right.right.left = TreeNode(4)
+    >>> t3.right.right.right = TreeNode(6)
+    >>> print(max_sum_bst(t3))
+    20
+    """
+    ans: int = 0
+
+    def solver(node: TreeNode | None) -> tuple[bool, int, int, int]:
+        """
+        Returns the maximum sum by making recursive calls
+        >>> t1 = TreeNode(1)
+        >>> print(solver(t1))
+        1
+        """
+        nonlocal ans
+
+        if not node:
+            return True, INT_MAX, INT_MIN, 0  # Valid BST, min, max, sum
+
+        is_left_valid, min_left, max_left, sum_left = solver(node.left)
+        is_right_valid, min_right, max_right, sum_right = solver(node.right)
+
+        if is_left_valid and is_right_valid and max_left < node.val < min_right:
+            total_sum = sum_left + sum_right + node.val
+            ans = max(ans, total_sum)
+            return True, min(min_left, node.val), max(max_right, node.val), total_sum
+
+        return False, -1, -1, -1  # Not a valid BST
+
+    solver(root)
+    return ans
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()