Maximum Width of Binary Tree in DSA C++ - Time & Space Complexity

Time Complexity: Maximum Width of Binary Tree

O(n)

Understanding Time Complexity

We want to know how the time needed to find the maximum width of a binary tree changes as the tree grows.

How does the number of nodes affect the work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int widthOfBinaryTree(TreeNode* root) {
    if (!root) return 0;
    unsigned long long maxWidth = 0;
    queue> q;
    q.push({root, 0});
    while (!q.empty()) {
        int size = q.size();
        unsigned long long start = q.front().second;
        unsigned long long end = q.back().second;
        maxWidth = max(maxWidth, end - start + 1);
        for (int i = 0; i < size; i++) {
            auto node = q.front().first;
            unsigned long long idx = q.front().second - start;
            q.pop();
            if (node->left) q.push({node->left, 2 * idx + 1});
            if (node->right) q.push({node->right, 2 * idx + 2});
        }
    }
    return (int)maxWidth;
}
    

This code finds the maximum width of a binary tree by level-order traversal, tracking positions to measure width.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop runs once per level of the tree, and inside it, a for loop processes all nodes at that level.
• How many times: Each node is visited exactly once during the traversal.

How Execution Grows With Input

As the number of nodes (n) grows, the code visits each node once, so the work grows directly with n.

Input Size (n)	Approx. Operations
10	About 10 visits and checks
100	About 100 visits and checks
1000	About 1000 visits and checks

Pattern observation: The operations increase linearly as the tree size increases.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the maximum width grows directly with the number of nodes in the tree.

Common Mistake

[X] Wrong: "The time depends on the height of the tree only because we process level by level."

[OK] Correct: Even though we go level by level, every node is visited once, so the total work depends on the total number of nodes, not just the height.

Interview Connect

Understanding how to analyze level-order traversal helps you explain many tree problems clearly and confidently in interviews.

Self-Check

"What if we used a depth-first search instead of a breadth-first search? How would the time complexity change?"