The triangles method to build X-trees from incomplete distance matrices

A method to infer X-trees (valued trees having X as set of leaves) from incomplete distance arrays (where some entries are uncertain or unknown) is described. It allows us to build an unrooted tree using only 2n-3 distance values between the n elements of X, if they fulfill some explicit conditions. This construction is based on the mapping between X-tree and a weighted generalized 2-tree spanning X. Résumé. On décrit une méthode pour la reconstruction des X-arbres (arbres valués admettant X comme ensemble de feuilles) à partir de tableaux de distances incomplets (où certaines valeurs sont incertaines ou inconnues). Elle permet de construire un arbre non orienté à partir de 2n-3 valeurs de distance entre les n éléments de X, sous des conditions qui sont explicitées. Cette construction est basée sur une relation entre X-arbres et 2-arbres valués généralisés d'ensemble de sommets X. Acknowledgements We are deeply grateful to Laurent Duret (L.G.B.P., Lyon) who is at the origin of this work and who helped us to improve the algorithm, providing partial distance matrices from his HOVERGEN database. We also thank an anonymous referee for his comments and suggestions.