Lee Mosher book definition of a tree.

Question

I was just reading the definition of a tree in Lee Mosher book, and he said if graph is simply connected then it is contractible.

I am wondering how is this true, can someone explain this to me please?Is there a proof for this fact?

Edit:

It seems like my question was not clear, here is what exactly I am asking, any simply connected graph is a tree,I know a priori that any tree is contractible.

Answer 1

Take a maximal spanning tree $T$ of the graph $\mathscr{G}$. You hopefully know that, by one possible definition, $T$ is contractible, and also that $\pi_1(\mathscr{G})$ is freely generated by all edges in $E(\mathscr{G})\setminus E(T)$. So if the graph is simply connected there can be no such edges, and $\mathscr{G}=T$ is contractible.

For details see Hatcher's discussion of fundamental groups of graphs.