If a binary tree is balanced, what is the maximum height of the tree in terms of
Practice Questions
Q1
If a binary tree is balanced, what is the maximum height of the tree in terms of the number of nodes 'n'?
O(n)
O(log n)
O(n log n)
O(1)
Questions & Step-by-Step Solutions
If a binary tree is balanced, what is the maximum height of the tree in terms of the number of nodes 'n'?
Step 1: Understand what a binary tree is. A binary tree is a structure where each node has at most two children.
Step 2: Know what it means for a binary tree to be balanced. A balanced binary tree is one where the height of the left and right subtrees of any node differ by at most one.
Step 3: Recognize that the height of a tree is the number of edges on the longest path from the root to a leaf.
Step 4: Realize that in a balanced binary tree, the nodes are arranged in such a way that they fill levels from top to bottom and left to right as much as possible.
Step 5: Understand that the maximum number of nodes in a balanced binary tree of height h is 2^(h+1) - 1.
Step 6: Rearranging this formula gives us h = log2(n + 1) - 1, which simplifies to O(log n) for large n.
Step 7: Conclude that in a balanced binary tree, the maximum height is O(log n), meaning it grows logarithmically with the number of nodes.
Balanced Binary Tree – A binary tree where the height of the left and right subtrees of any node differ by at most one, ensuring efficient operations.
Tree Height – The number of edges on the longest path from the root to a leaf node, which determines the efficiency of tree operations.
Big O Notation – A mathematical notation used to describe the upper limit of an algorithm's running time or space requirements in terms of input size.
