Calculate number of nodes from numbers of arcs in a tree - Induction

Question

If a tree has k arcs, how many nodes does it have?

Base case:

If k=0 => n=(k+1)=1

Inductive hypothesis: For every k, n=(k+1) is true

Proof:

Is it true for k=1?
k=n-1
1=n-1
1=(k+1)-1
k=1,so:
1=1+1-1
1=1
Proved?

Am I missing something?

Answer 1

You never want to use 0 or 1 as your base case for proof by induction

If k=0 => n=(k+1)=1

You can prove anything you want if you use 0 (or often 1) as your base case. The classic example of this being that all horses are the same color. This is your mistake.

In addition, you can make graphs with the same number of arcs have different numbers of nodes, which immediately shows that you are not going to be able to use proof by induction to solve your problem.

Good luck!