Given a tree $T\_1 = (V\_1, E\_1)$ (so a connected acyclic undirected graph is the definition I'm working with) and another tree $T\_2 = (V\_2, E\_2)$, is it true that given any node $v \in V\_1$ in $T\_1$ and $u \in V\_2$ in $T\_2$ $$T := ( V\_1 \cup V\_2, E\_1\cup \{v,u\}\cup E\_2)$$
Is still a tree? I am very convinced it is true, but I'm not able to prove it rigorously, could someone give me a hint or a counterexample if it's false indeed. Thank you.

Edit: $V\_1 \cap V\_2 = \emptyset$.