Finding the Minimum Depth of a Binary Tree

In this article, we’ll discuss how to determine the minimum depth of a binary tree, a common problem in tree traversal that has applications in various fields, such as database query optimization and decision-making algorithms. We'll explore an efficient solution using the Breadth-First Search (BFS) algorithm and explain why it’s optimal for this problem.

Problem Statement

Given a binary tree, the minimum depth is the shortest path from the root node to the nearest leaf node. A leaf node is defined as a node with no children.

For example, in the following binary tree, the minimum depth is 2:

The shortest path to a leaf node is 1 -> 3, which has a depth of 2.

      1
     / \
    2   3
   / \
  4   5        

Solution Approach

To solve this problem, we have two primary methods:

• Depth-First Search (DFS)
• Breadth-First Search (BFS)

While DFS is a viable approach, BFS is often more efficient for finding the minimum depth. This is because BFS explores the tree level by level, ensuring that we encounter the shallowest leaf node as soon as possible. This means we can stop searching as soon as we find the first leaf, which helps reduce unnecessary traversals.

Implementing the BFS Solution

In this solution, we use a queue to perform a level-order traversal of the binary tree. Here’s a step-by-step breakdown of our approach:

1. Initialize the Queue: Start by adding the root node to the queue along with a depth of 1, as we begin at the root level.
2. Traverse the Tree: Use a loop to process nodes in the queue:For each node, check if it’s a leaf. If it is, immediately return the current depth, as this is the minimum depth.If it’s not a leaf, add its children to the queue with an incremented depth.
3. Terminate on First Leaf: Since BFS processes each level fully before moving to the next, we’re guaranteed to find the minimum depth as soon as we reach the first leaf node.

Code Implementation

Here’s the Python code implementing the BFS approach:

Explanation

1. Edge Case: If the tree is empty (i.e., root is None), we return a depth of 0 since there are no nodes in the tree.
2. Queue Initialization: We start by creating a queue and adding the root node with a depth of 1.
3. Level Order Traversal: For each node, we check if it’s a leaf by verifying if it has no left or right child nodes. If it is a leaf, we return the current depth immediately. Since we’re using BFS, this ensures that we’re returning the minimum depth. If it’s not a leaf, we add its children to the queue with a depth incremented by 1.
4. Early Termination: As BFS explores each level fully before moving to the next, we find the minimum depth as soon as we encounter the first leaf node.

Complexity Analysis

• Time Complexity: O(N), where N is the total number of nodes in the tree. In the worst case, we may visit each node once.
• Space Complexity: O(N), as we store nodes in a queue for level-order traversal.

Why BFS is Optimal for Minimum Depth

While DFS is commonly used in tree problems, it’s more suitable when we want to explore each possible path, such as finding the maximum depth. For minimum depth, however, DFS may continue traversing deeper paths even after encountering a leaf node at a shallow depth, which adds unnecessary computations. BFS, on the other hand, ensures we stop at the shallowest leaf node, providing an optimal solution in terms of efficiency.