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Properties of Subtree-Prune-and-Regraft Operations on Totally-Ordered Phylogenetic Trees
Yun S. Song
Department of Computer Science, University of California, Davis, CA 95616, USA
yssong@cs.ucdavis.edu
Annals of Combinatorics 10 (1) p.147-163 March, 2006
AMS Subject Classification: 05C05, 92D15
Abstract:
We study some properties of subtree-prune-and-regraft (SPR) operations on leaflabelled rooted binary trees in which internal vertices are totally ordered. Since biological events occur with certain time ordering, sometimes such totally-ordered trees must be used to avoid possible contradictions in representing evolutionary histories of biological sequences. Compared to the case of plain leaf-labelled rooted binary trees where internal vertices are only partially ordered, SPR operations on totally-ordered trees are more constrained and therefore more difficult to study. In this paper, we investigate the unit-neighbourhood U(T), defined as the set of totally-ordered trees one SPR operation away from a given totally-ordered tree T. We construct a recursion relation for |U(T)| and thereby arrive at an efficient method of determining |U(T)|. In contrast to the case of plain rooted trees, where the unit-neighbourhood size grows quadratically with respect to the number n of leaves, for totally-ordered trees |U(T)| grows like O(n3). For some special topology types, we are able to obtain simple closed-form formulae for |U(T)|. Using these results, we find a sharp upper bound on |U(T)| and conjecture a formula for a sharp lower bound. Lastly, we study the diameter of the space of totally-ordered trees measured using the induced SPR-metric.
Keywords: SPR, ordered trees, neighbourhood, recombination

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