Structural theory of trees. I. Branching and condensations of trees

Abstract

Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of branching in trees and of condensed trees, which are trees in which every node is a branching node. We then introduce and investigate two different constructions of tree condensations – one shrinking, and the other expanding, the tree to a condensed tree.