Consider an amalgam $G \ast _A H$, or an HNN extension $G \ast_A$. For simplicity, suppose that $A = \{1\}$ or $A = \mathbb Z$. I'd like to use find the Bass-Serre tree associated with such splittings. I am aware of the construction of the universal cover of a graph of groups given in Serre [1], though I am struggling to put this construction into practice and examples seem to be lacking in the literature.
As a concrete example, consider the fundamental group of a torus $G = \pi_1(S^1 \times S^1) \cong \mathbb Z^2$, split along a simple closed curve $\{z\} \times S^1$ via the Seifert-van Kampen Theorem. This represents $G$ as an HNN extension $G \cong \mathbb Z \ast_{\mathbb Z}$. My question is, how can we explicitly construct the Bass-Serre tree for this splitting?
I am working on a problem where being able to construct explicit examples of such trees will prove very helpful, so any guidance on this would be appreciated. Thanks!
References
[1] Serre, J. P. (1980). Trees. Springer, Berlin, Heidelberg.