A virtually infinite cyclic group $G$ is quasi-isometric to $\mathbb{Z}$ and thus has two ends; by Stallings theorem, $G$ acts (without inversion) on a tree with finite edge-stabilizers.
But the construction of Stallings theorem works more generally for groups with more than one end; is it possible to find a simpler construction for virtually infinite cyclic groups?
There is a purely algebraic classification of the virtually infinite cyclic groups which can be reformulated as simple HNN extensions and amalgations over finite groups, so (using Bass-Serre theory) I hope that it is possible to construction a simple action of such a group on a tree.