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- Publisher
- Springer Journals
- Copyright
- Copyright © 2016 by The Author(s)
- Subject
- Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
- ISSN
- 0025-5858
- eISSN
- 1865-8784
- DOI
- 10.1007/s12188-016-0169-7
- Publisher site
- See Article on Publisher Site
Abstract
A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.
Journal
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Jan 4, 2017