A simplicial graph is a tree exactly when every two vertices are joined by a unique reduced path

If the graph is a simplicial tree, then [L1] gives connectedness, so every two vertices are joined by some reduced path. If there were two distinct reduced paths between the same vertices, follow one and return along the reverse of the other; after cancelling any common initial segment, one obtains a nontrivial reduced closed path, contradicting [L1].

Conversely, assume the graph is nonempty and every two vertices are joined by a unique reduced path. Then it is connected. A nontrivial reduced closed path based at $v_0$ with first edge $e_1$ would give two reduced paths from $v_0$ to $t(e_1)$: the one-edge path $e_1$ and the reverse of the remaining part of the cycle. That contradicts uniqueness. Hence no nontrivial reduced closed path exists, so the graph is a simplicial tree by [L1].