Answer:
To prove that the number of full vertices plus one is equal to the number of leaves in a non-empty binary tree, we can use mathematical induction.
Inductive Hypothesis:
Let's assume that the statement holds true for a binary tree with k vertices.
Base Case:
For a binary tree with a single vertex, we have one leaf and no full vertices. In this case, the number of full vertices plus one is equal to the number of leaves (0 + 1 = 1).
Inductive Step:
Now, let's assume that the statement holds true for a binary tree with k vertices. We will show that it also holds true for a binary tree with k+1 vertices.
Consider a binary tree with k+1 vertices. We can divide it into two parts: the left subtree and the right subtree. Each subtree is itself a binary tree with fewer vertices.
Let's denote the number of full vertices in the left subtree as L, and the number of full vertices in the right subtree as R. Also, let's denote the number of leaves in the left subtree as Ll and the number of leaves in the right subtree as Rl.
By the inductive hypothesis, the number of full vertices plus one in the left subtree is L+1, and the number of leaves in the left subtree is Ll.
Similarly, the number of full vertices plus one in the right subtree is R+1, and the number of leaves in the right subtree is Rl.
Now, let's consider the entire binary tree with k+1 vertices. There are two possible scenarios:
The root vertex has two children (full vertex):
In this case, the number of full vertices in the entire tree is L+R+1, and the number of leaves is Ll+Rl.
The root vertex has only one child (not a full vertex):
In this case, the number of full vertices in the entire tree is L+R, and the number of leaves is Ll+Rl+1.
In both scenarios, we observe that the number of full vertices plus one is equal to the number of leaves:
L+R+1 = Ll+Rl
L+R = Ll+Rl+1
Thus, by the principle of mathematical induction, we have proven that the number of full vertices plus one is equal to the number of leaves in a non-empty binary tree.
Explanation: