Generalised trees and Λ-trees

Summary

1. The idea of a Λ-tree, where Λ is an ordered abelian group, was introduced in [9]. We shall reproduce the definition shortly, but for an account of the basic theory of Λ-trees we refer to [1]. In the special case Λ = ℤ, ℤ-trees are closely related to simplicial trees (trees in the ordinary graph-theoretic sense). The connection is spelt out in Lemma 4 below, which shows that Λ-trees may be viewed as generalisations of simplicial trees. However, there are other notions of generalised tree in the literature, and our purpose here is to consider two of these, and their relation to Λ-trees.

Firstly there is what we call an order tree. This is a partially ordered set {P, ≤) such that the set of predecessors of any element is linearly ordered, that is, for all x, y, z ∈ P, if x ≤ z and y ≤ z, then either x ≤ y or y ≤ x. It is also convenient to assume that P has a least element (this can always be arranged just by adding one). By choosing a point in a Λ-tree, it is possible to make the Λ-tree into an order tree. We shall show that, conversely, any order tree (P, ≤) can be embedded in a Λ-tree for some suitable Λ, so that the ordering on P is induced from the ordering on the Λ-tree defined by the (image of) the least element of P.