On subgroups of finite complexity in groups acting on trees

Let G be a group acting on a tree X. We show that some classical results concerning finitely generated subgroups of free groups, free products, and free-by-finite groups, remain valid if we replace finitely generated subgroups by tame subgroups or by subgroups of finite complexity. We also prove new results for tame subgroups and, more generally, for subgroups of finite complexity.