I'm trying to find the Bass-Serre tree corresponding to the Baumslag-Solitar group $BS(1,2)$. I could find only one reference from the Internet, which I don't understand, and although I have the corresponding graph of groups, I cannot reconstruct this tree.
Edit: from the comments below, the corresponding tree would be an infinite binary tree but I'm missing the action of the group on that tree.
I believe it works like this:
Your tree $T$ is an infinite binary tree, where each vertex has one edge going in, and two edges going out.
Let your group be given by $G=\langle a,b\mid bab^{-1}=a^2\rangle$, and let it act on the right for this tree $T$, and pick a "starting vertex".
Find a copy of the real line in this tree, with the "starting vertex" at coordinate $0$. So at each vertex, $b$ sends some vertex along one of its children, left or right. $a$ acts by swapping these choices.
For example, if $b$ pushes the starting vertex along its left child, then $ab$ pushes it along its right child. $b^{-1}ab$ will send a vertex up to its parent, and down the other branch.
A nice set of pictures for this is in Meier's Groups, Graphs, and Trees. He doesn't discuss this action directly, but he shows how the Cayley graph of $G$ projects to the trivalent tree. It is pages 117--118 in my copy.