python coding test course, binary tree

A binary tree is one of the fundamental data structures in computer science and algorithms, playing a crucial role in many problems. Understanding binary trees and the ability to solve problems involving them are highly valued in coding interviews. In this article, we will select one problem related to binary trees and take a detailed look at the problem description and the solution process.

Problem: Maximum Depth of a Binary Tree

Write a function to find the maximum depth of a given binary tree. The depth of a binary tree is the number of nodes along the longest path from the root node down to the farthest leaf node. For example, let’s assume we have a binary tree as follows.

In this case, the maximum depth of the binary tree is 3 (node 1 → node 2 → node 4 or node 5). The signature of the function is as follows:

def maxDepth(root: TreeNode) -> int:

Problem Definition

The input parameter root given as input is the root node of the binary tree. This node is defined as an instance of the TreeNode class, which has pointers pointing to its left and right child nodes. If the binary tree is empty, the depth is 0.

Input Example

When calling maxDepth(root), the return value should be 3.

Output Example

Problem Solving Approach

A Depth-First Search (DFS) approach can be used to solve this problem. By using the DFS method to traverse the nodes of the tree, we can recursively calculate the depth from each node to its leaf nodes.

Step 1: Define the TreeNode Class

First, we need to write the TreeNode class that defines the nodes of the binary tree. Each node has a value and pointers to its left and right children.

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

Step 2: Define the Recursive Function for Maximum Depth

We will define a recursive function to calculate the maximum depth. We will use recursive calls to determine the depth of each subtree and select the maximum value among them.

def maxDepth(root: TreeNode) -> int:
    # Base case: If the node is None, the depth is 0
    if not root:
        return 0
    # Calculate the depth of the left and right subtrees
    left_depth = maxDepth(root.left)
    right_depth = maxDepth(root.right)
    # Return the maximum depth including the current node
    return max(left_depth, right_depth) + 1

Step 3: Implement the Final Function

We have now implemented the complete maxDepth function. This function returns the ‘maximum depth’ of the given binary tree.

Step 4: Analyze Time Complexity and Space Complexity

The time complexity of this algorithm is O(n), where n is the number of nodes in the binary tree. The time complexity is proportional to the size of the tree, as we visit each node once. The space complexity is O(h), where h is the height of the tree. In the worst case, the space complexity can be O(n), while in a balanced binary tree, it will be O(log n).

Test Cases

Let’s write some test cases to validate the function we created.

# Test cases 
def test_maxDepth():
    # Test case 1
    root1 = TreeNode(1, TreeNode(2, TreeNode(4), TreeNode(5)), TreeNode(3))
    assert maxDepth(root1) == 3, "Test case 1 failed"
    
    # Test case 2
    root2 = TreeNode(1)
    assert maxDepth(root2) == 1, "Test case 2 failed"
    
    # Test case 3: Empty tree
    root3 = None
    assert maxDepth(root3) == 0, "Test case 3 failed"
    
    # Test case 4
    root4 = TreeNode(1, TreeNode(2))
    assert maxDepth(root4) == 2, "Test case 4 failed"
    
    print("All test cases passed!")

test_maxDepth()

Conclusion

In this article, we introduced the problem of finding the maximum depth of a binary tree and explained the solution method in detail. There are many diverse problems related to binary trees, so it is important to practice by encountering various challenges. Problems like the Algorithm Challenge can help improve your skills further. Understanding the concept of binary trees and the basic principles of DFS traversal is greatly beneficial in coding tests. I hope you will continue to solve various algorithm problems to enhance your abilities.