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 U α of all elements of T whose order is α . T ∣ α is the union of all U β , β < α ; its length is α . A tree ( T 2 , ≤ 2 ) is called an extension of ( T 1 , ≤ 1 ) if ≤ 1 = ≤ 2 ∩ ( T 1 × T 1 ); T 2 is on end-extension of ( T 1 if T 1 = T 2 ∣ 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 .