Circular orders of tree metrics, and their uses for the reconstruction and fitting of phylogenetic trees

Abstract

The circular orders associated with the planar drawings of an X-tree (or phylogenetic tree) have been studied by several authors. They allow an encoding of an Xtree by 2n-3 numbers (where n is the number of elements of X), the lengths of some paths between leaves of the tree. It is shown here that circular orders are the same as those obtained from the table of a tree metric by a construction due to Yushmanov [28]. It is also observed that this construction applies to any dissimilarity, tree metric or not. Several fast algorithms (of complexity O(n2)) are derived from these results: for the determination of a Yushmanov order; for the reconstruction of the valued X-tree represention of a tree metric; for the recognition of a tree metric; and for the fitting of a tree metric to a given dissimilarity; the fitting method is based on successive local least squares approximations. Tested on various experimental and real data, it gives satisfactory results. RESUME. Plusieurs auteurs ont etudie les ordres circulaires associes aux representations planaires des X-arbres (ou arbres phylogenetiques), en particulier pour le codage d'un Xarbre value par 2n-3 longueurs de chemins d'une feuille a une autre (n etant le nombre d'elements de X). On montre dans cet article que ces ordres circulaires sont les memes que ceux obtenus a partir de la table d'une distance d'arbre par une construction due a Yushmanov [28]. De plus, cette construction s'applique a toute dissimilarite, distance d'arbre ou non. Ces resultats permettent d'obtenir plusieurs algorithmes rapides (de complexite O(n2)) : pour la determination d'un ordre de Yushmanov ; pour la reconstruction du X-arbre value representant une distance d'arbre ; pour la reconnaissance d'une distance d'arbre ; et pour l'ajustement d'une distance d'arbre a une dissimilarite d donnee. Ce dernier comporte des approximations locales successives par les moindres carres. Il conduit a une procedure d'ajustement qui donne des resultats interessants sur les exemples sur lesquels elle a ete testee. _______________ This research was partly supported by Esprit LTR Project n° 20244-ALCOM-IT. The authors are indebted to the Departement Informatique and the Centre Documentaire of the Ecole Nationale Superieure des Telecommunications de Paris for support in bibliographic research and programming work. They also wish to thank Alain Guenoche for helpful advice and comments.