Normalising graphs of groups.

We discuss a partial normalisation of a finite graph of finite groups $$(\Gamma (-), X)$$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include: (1) an important inequality for the number of edges in a Stallings decomposition $$\Gamma \cong \pi _1(\Gamma (-), X)$$ of a finitely generated virtually free group, (2) the proof of equivalence of a number of conditions for such a group to be 'large', as well as (3) the classification up to isomorphism of virtually free groups of (free) rank 2. We also discuss some number-theoretic consequences of the last result. [ABSTRACT FROM AUTHOR]