Tree sets

TL;DR

This paper introduces the concept of tree sets, an abstract framework capturing the core structure of various tree-like combinatorial objects, including infinite graphs and matroids, unifying their formalization.

Contribution

It formalizes tree sets as a unifying abstraction for diverse tree structures, extending applicability to infinite and more complex combinatorial objects.

Findings

Tree sets encompass various tree-like structures in combinatorics.

Order trees are represented as oriented tree sets.

The paper details how different structures can be reconstructed from tree sets.

Abstract

We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and matroids etc. Unlike graph-theoretical or order trees, these _tree sets_ can provide a suitable formalization of tree structure also for infinite graphs, matroids, and set partitions. Order trees reappear as oriented tree sets. We show how each of the above structures defines a tree set, and which additional information, if any, is needed to reconstruct it from this tree set.