Properties of Subtree-Prune-and-Regraft Operations on Totally-Ordered Phylogenetic Trees
نویسنده
چکیده
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.
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