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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?

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  • $\begingroup$ Did you mean "Assume $G$ is connected?" $\endgroup$ – angryavian Apr 11 at 0:32
  • $\begingroup$ @angryavian Yes correct! sorry about that $\endgroup$ – Hidaw Apr 11 at 0:35
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You would need more than just there be a connected graph on $uw$ and a connected graph on $wv.$ Here is a way I would go about it.

Since every vertex of $G$ is linked by a path, $G$ is connected. If $G$ had a cycle, and $u,w$ were on that cycle, then they would be connected by two distinct paths. This cannot happen, so $G$ has no cycles. The only connected acyclic graphs are trees, hence $G$ is a tree.

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