What is the maximum number of nodes in a binary tree of height h?
Practice Questions
Q1
What is the maximum number of nodes in a binary tree of height h?
h
2^h - 1
2^h
h^2
Questions & Step-by-Step Solutions
What is the maximum number of nodes in a binary tree of height h?
Step 1: Understand what a binary tree is. A binary tree is a tree data structure where each node has at most two children, referred to as the left child and the right child.
Step 2: Know what height means. The height of a binary tree is the number of edges on the longest path from the root node to a leaf node. If a tree has only one node (the root), its height is 0.
Step 3: Recognize that at each level of the tree, the number of nodes can double. For example, at height 0 (the root), there is 1 node. At height 1, there can be 2 nodes. At height 2, there can be 4 nodes, and so on.
Step 4: Realize that the maximum number of nodes at each height can be calculated using powers of 2. Specifically, at height h, the maximum number of nodes is 2^h.
Step 5: To find the total maximum number of nodes in the entire tree up to height h, you need to add the nodes from all levels: 1 (at height 0) + 2 (at height 1) + 4 (at height 2) + ... + 2^h (at height h).
Step 6: The sum of nodes from height 0 to height h can be calculated using the formula for the sum of a geometric series, which results in 2^(h+1) - 1.
Step 7: Therefore, the maximum number of nodes in a binary tree of height h is 2^(h+1) - 1, which simplifies to 2^h - 1 when considering the total nodes from height 0 to h.
