Final answer:
The number of levels in a perfect binary tree can be proven using mathematical induction. We verify the base case for k=1, and then assume the formula n = 2^k - 1 is true for k levels to prove it for k+1 levels. Option A, Mathematical Induction, is the correct method for this proof.
Step-by-step explanation:
Proof of the Number of Levels in a Perfect Binary Tree
To prove the number of levels in a perfect binary tree, we often use mathematical induction. A perfect binary tree is a binary tree in which all interior nodes have exactly two children and all leaves are at the same level or depth. To start, we will define some terms: let 'n' denote the total number of nodes in the perfect binary tree, and 'k' denote the total number of levels. The relationship between the total number of nodes and the levels in a perfect binary tree is given by the formula: n = 2k - 1.
Mathematical induction works in two steps: the base case and the inductive step. For the base case, let's consider a tree with just one level (k = 1). This means there's only one node, which matches our formula since 21 - 1 = 1. So, our base case holds true.
For the inductive step, we assume that our formula holds true for a tree with 'k' levels. So, if a tree has k levels, it has 2k - 1 nodes. Now let's add one more level to the tree, making it 'k+1' levels. Each level in a binary tree doubles the maximum number of nodes possible, so we can calculate the nodes for 'k+1' levels as 2 * (2k - 1) + 1, which simplifies to 2k+1 - 1, this upholds the original relationship in our formula for a tree with 'k+1' levels, completing our inductive step.
Therefore, the correct option for this proof is A. Mathematical Induction.