On some relations between 2-trees and tree metrics

Abstract A tree function (TF) t on a finite set X is a real function on the set of the pairs of elements of X satisfying the four-point condition: for all distinct x, y, z, w ∈ X, t(xy)+t(zw)⩽ max{t(xz) + t(yw), t(xw) + t(yz)}. Equivalently, t is representable by the lengths of the paths between the leaves of a valued tree Tl. TFs are a straightforward generalization of the tree dissimilarities and tree metrics of the literature. A graph Θ is a 2-tree if it belongs to the following class Q : an edge-graph belongs to Q : if Θ′ ∈ Q and yz is an edge of Θ′, then the graph obtained by the addition to Θl of a new vertex x adjacent to y and z belongs to Q . These graphs, and the more general k-trees, have been studied in the literature as generalizations of trees. It is first explicited here how to make a TF tΘ, d correspond to any positively valued 2-tree Θd on X. Then, given a tree dissimilarity t, the set Q(t) of the 2-trees Θ such that t = tΘ, t is studied. Any element of Q(t) gives a way of summarizing t by its restriction to a minimal subset of entries. Several characterizations and properties of the elements of Q(t) are given. We describe five classes of such elements, including two new ones. Associated with a dissimilarity of the general type, these classes of 2-trees lead to methods for the recognition and fitting of tree dissimilarities.

[1]  Alain Guénoche,et al.  Trees and proximity representations , 1991, Wiley-Interscience series in discrete mathematics and optimization.

[2]  Bruno Leclerc,et al.  Minimum spanning trees for tree metrics: abridgements and adjustments , 1995 .

[3]  F. Harary,et al.  On acyclic simplicial complexes , 1968 .

[4]  A. Dobson.  Unrooted trees for numerical taxonomy , 1974, Journal of Applied Probability.

[5]  M. Schader,et al.  New Approaches in Classification and Data Analysis , 1994 .

[6]  D. Kendall,et al.  Mathematics in the Archaeological and Historical Sciences , 1971, The Mathematical Gazette.

[7]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[8]  Bruno Leclerc.  Minimum Spanning Trees and Types of Dissimilarities , 1996, Eur. J. Comb..

[9]  S. Chaiken,et al.  An Optimal Diagonal Tree Code , 1983 .

[10]  Arnold G. Kluge,et al.  A Numerical Approach to Phylogenetic Systematics , 1970 .

[11]  Hans-Jürgen Bandelt.  Recognition of Tree Metrics , 1990, SIAM J. Discret. Math..

[12]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[13]  Lowell W. Beineke,et al.  Characterizations of 2-dimensional trees , 1969 .

[14]  B. Leclerc.  Description combinatoire des ultramétriques , 1981 .

[15]  S. F. Kapoor,et al.  The Many Facets of Graph Theory , 1969 .

[16]  H. Colonius,et al.  Tree structures for proximity data , 1981 .

[17]  L. Hubert.  SPANNING TREES AND ASPECTS OF CLUSTERING , 1974 .

[18]  Frank Critchley.  On exchangeability-based equivalence relations induced by strongly Robinson and, in particular, by quadripolar Robinson dissimilarity matrices , 1994 .

[19]  Bruno Leclerc,et al.  Adjustments of tree metrics based on minimum spanning trees , 1994 .

[20]  F. Roush.  Les arbres et les representations des proximites : J.-P. Barthelemy and A. Guenoche, Paris: Masson, 1988, 236 pages, 160 francs. , 1989 .

[21]  Lowell W. Beineke,et al.  The number of labeled k-dimensional trees , 1969 .

[22]  Hans-Jürgen Bandelt,et al.  Symmetric Matrices Representable by Weighted Trees Over a Cancellative Abelian Monoid , 1995, SIAM J. Discret. Math..

[23]  P. Buneman.  The Recovery of Trees from Measures of Dissimilarity , 1971 .

[24]  Bruno Leclerc,et al.  The residuation model for the ordinal construction of dissimilarities and other valued objects , 1994 .

[25]  S. Hakimi,et al.  The distance matrix of a graph and its tree realization , 1972 .

[26]  J. Moon.  The number of labeled k-trees , 1969 .

[27]  R. Prim.  Shortest connection networks and some generalizations , 1957 .