# Binary tree proof

# Recall that a binary tree can be defined recursively as:

* A Binary Tree is either empty

* or A Binary Tree consists of a node with a left and right child both of which are Binary Trees.

The degree of a node in a tree is equal to 0 if both children are empty, 1 if one of the children are empty, and 2 of both children are not empty. Use induction to show that the number of nodes in a binary tree is equal to one more than the sum of the degrees of the nodes in a binary tree.

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#### Solution Preview

Proof:

We use induction for the number of nodes n in a binary tree. We assume S(n) is the sum of the degrees of n nodes in a binary tree. We want to show n=S(n)+1.

When n=1, then the binary tree has only one nodes. This node has no children, so the degree of this node is 0. Thus S(1)=0. So we have 1=S(1)+1.

When n=2, then the binary tree has ...

#### Solution Summary

The solution provides a proof regarding binary trees and nodes.