Order (2018) 35:171–192
Received: 21 December 2015 / Accepted: 28 February 2017 / Published online: 25 April 2017
© The Author(s) 2017.
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 formal-
ization 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.
Keywords Tree · Order · Nested · Graph · Matroid · Protree
There are a number of concepts in combinatorics that express the tree-likeness of discrete
structures. Among these are:
nested subsets, or bipartitions, of a set.
Other notions of tree-likeness, such as tree-decompositions of graphs or matroids, are
modelled on these.
There are also non-discrete such concepts, such as R-trees, which are not our topic here.
Hamburg University, Hamburg, Germany