A graph of groups construction is a generalisation of free products of groups, and it also generalises another construction called an HNN extension. The topological viewpoint gives rise to an action of the constructed group on a tree, and conversely a group action on a tree leads to a graph of groups. This theory was developed by Jean-Pierre Serre and Hyman Bass to study some linear algebraic groups in J-P Serre's monograph `Arbres, amalgames, $\mathrm {SL}_2$' (title of the English translation: `Trees'), but quickly became a standard tool in geometric topology and geometric group theory.
In this talk, I will present the correspondence between graphs of groups and groups acting on trees.