To solve this problem and prove that every complete binary tree with height $h$ must have $2^h$ leaves, let us proceed step-by-step:

Definitions and Understanding

1. A complete binary tree is a binary tree where all levels except possibly the last are fully filled, and all nodes in the last level are as far left as possible.
2. Height $h$ of a binary tree is the number of edges from the root to the deepest leaf.

Prove $2^h$ Leaves

We will use induction to prove the given statement.

Base Case:

For $h = 0$:

• A binary tree with height $0$ has only a root node, which is also the only leaf. The number of leaves is $2^0 = 1$, which holds true.

Inductive Step:

Assume that a complete binary tree of height $h-1$ has $2^{h-1}$ leaves. This is the inductive hypothesis.

Now, for a complete binary tree of height $h$:

• In a complete binary tree, every level is filled except possibly the last one.
• The last level will have nodes as left-packed as possible, meaning the tree grows from $2^0$ nodes at the root, $2^1$ at the next level, and so on.
• Therefore, the maximum number of nodes at the last level (level $h$) is $2^{h}$, since it doubles at each level.
• As a result, the total number of leaves in the tree when it is complete is $2^h$.

By the inductive hypothesis, the number of leaves is $2^h$, proving the statement.

Would you like further details or clarifications on the steps? Here are some additional related questions for practice:

1. Can you derive the total number of nodes in a complete binary tree of height $h$?
2. How does the definition of a complete binary tree differ from a full binary tree?
3. Can a binary tree with height $h$ have fewer than $2^h$ leaves? Explain why or why not.
4. What is the relationship between the number of internal nodes and leaves in a complete binary tree?
5. How does the proof change if the binary tree is not complete but balanced?

Tip: Induction is a powerful method to prove statements in recursive structures like trees. Always clearly define the base case and hypothesis.