I am thinking about something along these lines:
Assume that $G$ is connected and let $u$ and $v$ be two vertices that are disjoint. In order for $u$ and $v$ to be connected there exists another vertex $w$ such that $uw$ and $wv$ are connected. Now since $u$ and $v$ are both connected with $w$ then $G$ is a tree.
Does that work OR is it more complicated than that?