Introduction

The idea of a protree is due to M.J. Dunwoody, and they first appear in [7], although the name protree was not used until later, in [8]. Two major advances in combinatorial group theory in the late 1960's were the Bass-Serre theory (see [10]), and Stallings’ work on ends of groups (there is an account of this work in [11], and from a different perspective in [5]; a more recent and more general account can be found in [6]). Together, these imply that a finitely generated group with more than one end acts on a tree with finite edge stabilisers. This raised the problem of giving a direct construction of the tree, and it was in solving this problem that Dunwoody introduced protrees. Under certain circumstances (the finite interval condition, which will be considered in §3 below), a protree gives rise to an ordinary simplicial tree.

We show here that any protree arises in a simple way from a Λ-tree, for some suitable ordered abelian group Λ. For information on Λ-trees, see [1]. This is part of a programme, started in [4], to demonstrate that any notion of generalised tree which occurs in the literature is a manifestation of some suitable Λ-tree.