Introduction:

In the realm of Binary Search Trees (BSTs), a critical concern is identifying dead ends, which occur at leaf nodes where no further insertion is feasible without violating the BST property. This problem requires traversing the BST and discerning nodes that could potentially lead to a dead end. Let’s delve into a solution using Python to identify and resolve these dead ends efficiently.

Problem Statement:

Given a Binary Search Tree (BST) containing unique positive integers greater than 0, the task is to devise a function isDeadEnd that identifies whether the BST contains a dead end or not. A dead end occurs when a leaf node prohibits the insertion of any additional nodes.

Example Scenarios:

Consider two scenarios to illustrate the problem:

1. Scenario 1:

               8
             /   \ 
           5      9
         /  \     
        2    7 
       /
      1

Output: Yes

Explanation: Node 1 represents a Dead End in the given BST.

2. Scenario 2:

              8
            /   \ 
           7     10
         /      /   \
        2      9     13

Output: Yes

Explanation: Node 9 forms a Dead End in the provided BST.

Solution Approach (Python):

class Solution:
    def isDeadEnd(self, root):
        def isDeadEndUtil(node, min_val, max_val):
            if not node:
                return False

            if min_val == max_val:
                return True

            left = isDeadEndUtil(node.left, min_val, node.data - 1)
            right = isDeadEndUtil(node.right, node.data + 1, max_val)

            return left or right

        return isDeadEndUtil(root, 1, float('inf'))

Explanation of the Solution:

The solution employs a recursive helper function isDeadEndUtil. This function accepts the current node, along with the minimum and maximum allowable values within the current subtree. It examines the minimum and maximum values to identify potential dead ends and explores the left and right subtrees recursively.

The isDeadEnd function initializes the recursive helper function with the root node and the minimum and maximum possible values for BST nodes, which are 1 and positive infinity, respectively.

Complexity Analysis:

• Time Complexity: O(N), where N is the number of nodes in the BST.
• Space Complexity: O(N) due to recursive function calls and stack space for recursion.

This solution effectively detects dead ends within a BST, enabling the identification of leaf nodes that inhibit further insertions in the tree structure.

Conclusion:

In the realm of Binary Search Trees (BSTs), the challenge of identifying dead ends—leaf nodes that restrict further insertions—poses a significant concern. The provided Python solution efficiently checks whether a BST contains a dead end by traversing the tree and analyzing node values, enabling the detection of potential limitations in tree expansion.