We use Nielsen methods to study generating sets of subgroups of groups that act on simplicial trees and give several applications. In particular, we exhibit an explicit bound for the complexity of acylindrical splittings of a finitely generated group in terms of its rank. This is applied to JSJ-splittings of word-hyperbolic groups and 3-manifolds. As a last application we construct examples of amalgamated products that show that there exists no non-trivial rank formula for amalgamated products.

2000 Mathematical Subject Classification: 20E06, 20E08, 57M27.