Journal of Mathematical Biology

, Volume 61, Issue 2, pp 253–276 | Cite as

Nodal distances for rooted phylogenetic trees

  • Gabriel Cardona
  • Mercè Llabrés
  • Francesc Rosselló
  • Gabriel Valiente


Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. For rooted phylogenetic trees, however, these vectors can only separate non-weighted binary trees, and therefore these dissimilarity measures are metrics only on this class of rooted phylogenetic trees. In this paper we overcome this problem, by splitting in a suitable way each path length between two taxa into two lengths. We prove that the resulting splitted path lengths matrices single out arbitrary rooted phylogenetic trees with nested taxa and arcs weighted in the set of positive real numbers. This allows the definition of metrics on this general class of rooted phylogenetic trees by comparing these matrices through metrics in spaces \({\mathcal{M}_n(\mathbb {R})}\) of real-valued n × n matrices. We conclude this paper by establishing some basic facts about the metrics for non-weighted phylogenetic trees defined in this way using L p metrics on \({\mathcal{M}_n(\mathbb {R})}\), with \({p \in \mathbb {R}_{ >0 }}\).


Phylogenetic tree Weighted tree Nested taxa Path lengths Nodal metric 

Mathematics Subject Classification (2000)

92B10 92D15 05C05 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Gabriel Cardona
    • 1
  • Mercè Llabrés
    • 1
    • 2
  • Francesc Rosselló
    • 1
    • 2
  • Gabriel Valiente
    • 2
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of the Balearic IslandsPalma de MallorcaSpain
  2. 2.Research Institute of Health Science (IUNICS)Palma de MallorcaSpain
  3. 3.Algorithms, Bioinformatics, Complexity and Formal Methods Research GroupTechnical University of CataloniaBarcelonaSpain

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