Annals of Combinatorics

, Volume 10, Issue 1, pp 147–163

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

Original Paper

DOI: 10.1007/s00026-006-0279-5

Cite this article as:
Song, Y.S. Ann. Comb. (2006) 10: 147. doi:10.1007/s00026-006-0279-5


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.


SPRordered treesneighbourhoodrecombination

AMS Subject Classification.


Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CaliforniaDavisUSA