Annals of Combinatorics

, Volume 17, Issue 1, pp 205–228

Affine and Projective Tree Metric Theorems


DOI: 10.1007/s00026-012-0173-2

Cite this article as:
Kleinman, A., Harel, M. & Pachter, L. Ann. Comb. (2013) 17: 205. doi:10.1007/s00026-012-0173-2


The tree metric theorem provides a combinatorial four-point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A related weaker four point condition characterizes dissimilarity maps derived from circular split systems known as Kalmanson metrics. The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics, and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.


hierarchy Gromov product Kalmanson metric Robinsonian metric PC-tree PQ-tree phylogenetics pyramid ultrametric 

Mathematics Subject Classification

05C05 92B10 

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsCourant Institute of Mathematical SciencesNew YorkUSA
  3. 3.Departments of Mathematics and Molecular & Cell BiologyUniversity of CaliforniaBerkeleyUSA