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

TL;DR

This paper develops a structural theory of trees focusing on branching and condensation, introducing new concepts and constructions for condensed trees, which are trees where every node is a branching node.

Contribution

It introduces and explores notions of branching and condensed trees, along with two methods for constructing condensed trees through shrinking and expanding processes.

Findings

Defined and analyzed various notions of branching in trees.

Introduced two methods for constructing condensed trees.

Provided foundational results on the structure of condensed 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 \emph{branching in trees} and of \emph{condensed trees}, which are trees in which every node is a branching node. We then introduce and investigate two different constructions of \emph{tree condensations} -- one shrinking, and the other expanding, the tree to a condensed tree.