[Math] Prove that if $T$ has only vertices of degree 1, 2 and 3, and $T$ has exactly 10 vertices of degree 3, then $T$ has 12 leaves.

Question

Here is the original problem:

Consider a tree $T$ that has only vertices of degree 1, 2 and 3. Suppose that $T$ has exactly 10 vertices of degree 3. Find and prove how many leaves $T$ has.

My Thoughts

I know that any tree $T$ satisfying those conditions has 12 leaves, but I'm not really sure how to prove it. I believe you first have to start with a vertex. Then, place three vertices around this vertex to make a vertex of degree 3. Repeat this step, making sure the graph is a tree (no cycles and loops).

Any comments or thoughts?

Answer 1

Hint. The number of vertices of degree $2$ is irrelevant. Can you see why?

So, assume there are no vertices of degree $2$, then follow the ideas other people have suggested.