Show that a acylcic graph with $2n$ vertices $n-1$ of which are of degree 3 and the rest $n + 1$ are of degree 1 is a tree.

Question

Show that a acylcic graph with $2n$ vertices $n-1$ of which are of degree 3 and the rest $n + 1$ are of degree 1 is a tree.

My attempt: I was trying to show that if a acylcic graph have an n-1 edges it's connected, but I couldn't prove it.

Any tips?

Answer 1

Asume that the graph is not connected.

Let $A_1$ and $A_2$ be the connected components. Both should be acylcic ($G$ is asyclic). Let $A_1$ have $k$ vertices.

Then $A_1$ should have $k-1$ edges. Then $A_2$ have $n - k$ vertices and $n - k$ edges. Hence $A_2$ is not acyclic which contradicts with G being acyclic.