Trees

Abstract

This is an attempt to give a survey of recent results concerning trees. The article is an extended version of our talk in Oberwolfach (Schwarzwald) last May; the forests surrounding the Forschungsinstitut turned out to be a good inspiration.A tree is a partially ordered set T = (T, ≤) such that for every x ∈ T, the set = {y ∈ T: y < x} is well-ordered. The order type of is called the order of x, o(x), and the length of T is sup {o(x) + 1: x ∈ T}; an α-tree (where α is an ordinal) is a tree of length α. The αth level of T is the set Uα of all elements of T whose order is α. T∣α is the union of all Uβ, β < α; its length is α. A tree (T2, ≤2) is called an extension of (T1, ≤1) if ≤1 = ≤2 ∩ (T1 × T1); T2 is on end-extension of (T1 if T1 = T2∣α for some α. A maximal linearly ordered subset of a tree T is called a branch of T; an α-branch is a branch of length α.