Maximum Depth of Binary Tree

Clarify the problem:

• The problem requires finding the maximum depth of a binary tree.
• We need to implement a function that takes the root of a binary tree as input and returns the maximum depth of the tree.

Analyze the problem:

• Input: The root node of a binary tree.
• Output: An integer representing the maximum depth of the binary tree.
• Constraints:
  • The binary tree can be empty (null), in which case the maximum depth is 0.
  • The binary tree can have multiple levels.

Design an algorithm:

• We can solve this problem using a recursive approach.
• If the root is null, we return 0 since there are no nodes in the tree.
• Otherwise, we recursively calculate the maximum depth of the left and right subtrees.
• The maximum depth of the entire tree is the maximum depth of the subtrees plus 1.

Explain your approach:

• We will implement a function called maxDepth that takes the root of a binary tree as input.
• If the root is null, we return 0.
• Otherwise, we recursively calculate the maximum depth of the left and right subtrees.
• We return the maximum of the depths of the subtrees plus 1.

Write clean and readable code:

python

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

def maxDepth(root):
    if root is None:
        return 0
    left_depth = maxDepth(root.left)
    right_depth = maxDepth(root.right)
    return max(left_depth, right_depth) + 1

Test your code:

python

# Test case 1
# Input:
#     3
#    / \
#   9  20
#     /  \
#    15   7
# The binary tree has a maximum depth of 3.
# Expected output: 3
root = TreeNode(3)
root.left = TreeNode(9)
root.right = TreeNode(20)
root.right.left = TreeNode(15)
root.right.right = TreeNode(7)
assert maxDepth(root) == 3

# Test case 2
# Input: null
# An empty tree has a maximum depth of 0.
# Expected output: 0
assert maxDepth(None) == 0

Optimize if necessary:

• The current solution has a time complexity of O(n), where n is the number of nodes in the binary tree.
• This is because we visit each node once during the depth-first search.
• The space complexity is O(h), where h is the height of the binary tree.
• This is because the maximum depth of the recursion stack is equal to the height of the tree.

Handle error cases:

• The code assumes that the input is a valid binary tree.
• It does not handle invalid input or error cases, such as non-tree inputs.
• In such cases, the behavior of the code is undefined.

Discuss complexity analysis:

• The time complexity of the solution is O(n), where n is the number of nodes in the binary tree.
• The space complexity is O(h), where h is the height of the binary tree.
• The best-case scenario occurs when the binary tree is empty, and the function returns 0 in constant time.
• The worst-case scenario occurs when the binary tree is a skewed tree, and the function traverses all n nodes, resulting in O(n) time complexity.
• The average-case time complexity is also O(n) for random inputs.
• The space complexity depends on the height of the binary tree, and it can be O(n) in the worst case for a skewed tree.