I want to use Euler’s theorem for planar graphs to proof that for a tree $T = (V, E)$ that $|V | = |E| + 1$.
Now It's very obvious that a tree is a planar graph since it is connected and there is no edge intersections. So we can apply Euler's theorem which states that for any planar connected graph we have that $|V| + |F| = |E| + 2$, Now we know that the tree itself will contribute one face to the number of faces in a tree, and in fact it's the only face in a tree, since all other faces should be cycles, and there exists no cycles in a tree and so $|F| = 1$ for any tree. And so $$|V| +1 = e + 2 \implies |V| = |E| + 1$$.
I was wondering If this argument is correct, Is it correct to say that there exists only one face for any tree which is basically the whole tree itself and there exists no other faces since there is no cycles in a tree by definition ?
This is correct. A tree has no cycles, so no closed regions, so there is only one face-the outer region.