Connection between two trees

Question

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: $V_1 \cap V_2 = \emptyset$.

Answer 1

As removing any edge from a tree results in exactly two trees, your conjecture is true (for disjoint trees).

Furthermore, graph A has no cycles, and nor does graph B. A cycle must therefore travel from graph A to graph B and back again, but we have only supplied one new edge to do this with.

Answer 2

Hint: If the vertex sets $V_1,V_2$ are disjoint, then adding an edge $\{u,v\}$ between the two trees makes the composite graph a tree. Prove acyclicity and connectedness.

Answer 3

Counterexample:

$$V_1=V_2=\{1,2,3\}, E_1=\{\{1,2\},\{1,3\}\}, E_2=\{\{1,2\},\{2,3\}\}, u=2, v=3$$