Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6423588 | Discrete Mathematics | 2011 | 5 Pages |
We provide a proof of Sholander's claim [M. Sholander, Trees, lattices, order, and betweenness, Proceedings of the American Mathematical Society 3 (1952) 369-381] 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 [L. Burigana, Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185 (2009) 5-36] of tree betweenness and provide a relatively short proof.
⺠Finite Sholander trees are trees in the usual sense. ⺠This yields a new axiomatic characterization of the interval function of a tree. ⺠We give a short proof of Burigana's axiomatic characterization of tree betweenness.