Solution Explanation

This problem asks to find the k closest values in a Binary Search Tree (BST) to a given target value. The solution utilizes a depth-first search (DFS) approach with a queue (or deque in Python) to efficiently maintain the k closest values found so far.

Approach

1. DFS Traversal: The solution employs an in-order DFS traversal of the BST. In-order traversal ensures that the nodes are visited in ascending order of their values. This is crucial because it allows us to efficiently keep track of the closest values as we traverse the tree.

2. Queue (or Deque): A queue (or deque in Python, which provides O(1) time complexity for both append and popleft) is used to store the k closest values encountered during the traversal.

3. Maintaining Closeness: For each node visited, its distance from the target is calculated. If the queue has fewer than k elements, the node's value is added to the queue. If the queue is full (contains k elements), and the current node is closer to the target than the furthest element in the queue (the element at the head/front), the furthest element is removed from the queue, and the current node's value is added. This ensures that the queue always contains the k closest values encountered so far.

4. Return Result: After the DFS traversal completes, the queue (or deque) contains the k closest values to the target. The contents of the queue are converted into a list and returned as the solution.

Time and Space Complexity Analysis

• Time Complexity: The time complexity is O(N) in the worst case, where N is the number of nodes in the BST. This happens because in the worst case, we might have to visit all nodes in the tree. However, if the tree is balanced and the target is near the middle, the number of nodes visited would be significantly smaller. The operations within the DFS (comparison, queue operations) are all O(1).

• Space Complexity: The space complexity is O(H + k) in the worst case, where H is the height of the BST. O(H) comes from the recursive call stack during DFS, and O(k) is due to the queue storing k elements. In a balanced BST, H = log₂N, so the space complexity becomes O(log₂N + k). In the worst-case scenario of a skewed BST, H = N.

Code Explanation (Python)

The Python code utilizes a collections.deque for efficient queue operations. The dfs function recursively traverses the tree. The logic to maintain the k closest elements in the deque is implemented as described in the approach section. Finally, the deque is converted to a list before returning.

Code Explanation (Java, C++, Go)

The Java, C++, and Go code follow a similar structure to the Python code. They use a LinkedList (Java), a queue (C++), and a slice (Go) to manage the k closest elements. The core logic remains the same: a recursive DFS traversal, calculating distances to the target, and maintaining a queue of the closest k values seen so far. The difference lies primarily in the syntax and data structures used in each language.