In Hatcher on page 84 there is the following proposition: For a connected graph $X$ with maximal tree $T$, $\pi_1 (X)$ is a free group with basis the classes $[f_\alpha]$ corresponding to the edges $e_\alpha$ of $X - T$.
I tried to apply this to the torus $T^2$ with the two edges $e_a, e_b$ and the maximal tree $T = \{ x_0\}$ where $x_0$ is the point connecting the two edges.
The problem is that then I get the free group $F(a,b)$ instead of $\mathbb{Z} \oplus \mathbb{Z}$.
Where am I making the mistake? Many thanks for your help!