I'm currently reading the book Trees by J.P Serre. I'm interested in the results of amalgamated groups given in chapter 1, section 4 aptly named Trees and amalgams. One of the first results he proves is that any action of a group $G$ on a tree $T$ has a fundamental domain iff $G/T$ is a tree.
I was then trying to think of an example of a group $G$ and a tree $T$ such that the quotient $G/T$ is not a tree to better understand this. But every action I can think on a given tree is such that the quotient ends up being a subgraph, that has to be connected, thus a tree. Actually I don't have a big array of actions of groups on trees test this so I would gladly accept examples.
Consider the action of $\mathbb{Z}$ on the line $\mathbb{R}$, considered as a tree with each integer point a vertex, by translation. Then the quotient is a loop (since all vertices are identified), and so is not a tree. There's no strict fundamental domain since fundamental domains have to be closed, and strict ones (as I believe Serre means) can only contain one point for each orbit, yet any closed set containing an entire open interval has to contain both endpoints.