What is the height of a balanced binary tree with n nodes?
Practice Questions
Q1
What is the height of a balanced binary tree with n nodes?
O(n)
O(log n)
O(n log n)
O(1)
Questions & Step-by-Step Solutions
What is the height of a balanced binary tree with n nodes?
Step 1: Understand what a balanced binary tree is. A balanced binary tree is a tree where the height of the left and right subtrees of any node differ by at most one.
Step 2: Know that the height of a tree is the number of edges on the longest path from the root to a leaf.
Step 3: Realize that in a balanced binary tree, the number of nodes increases exponentially as the height increases. Specifically, a balanced binary tree with height h can have at most 2^(h+1) - 1 nodes.
Step 4: If we have n nodes, we can set up the equation 2^(h+1) - 1 >= n to find the maximum height h.
Step 5: Rearranging the equation gives us 2^(h+1) >= n + 1. Taking the logarithm base 2 of both sides gives us h + 1 >= log2(n + 1).
Step 6: Therefore, h >= log2(n + 1) - 1. This means the height h is approximately O(log n) because we can ignore the constant and lower order terms.
