We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions. Much of our analysis deals with conditions on an action of a group $Gamma$ on a tree $T$ which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree $T$ and on its boundary $partial T$, and faithfulness in a strong sense. For this, we define in particular the notion of a emph{slender} automorphism of $T$, namely of an automorphism such that its set of fixed points on $partial T$ is nowhere dense with respect to the shadow topology.