Finite Sholander Trees, Trees, and their Betweenness

TL;DR

This paper proves a key claim about representing segments with trees, characterizes the interval function of a tree, and simplifies existing characterizations of tree betweenness relations.

Contribution

It provides a proof of Sholander's claim on segment representation and streamlines Burigana's characterization of tree betweenness relations.

Findings

Proof of Sholander's claim on segment representation

Characterization of the interval function of a tree

Simplified proof of Burigana's characterization

Abstract

We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof.