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.
KeywordsTree Order Nested Graph Matroid Protree
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