<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 11 Issue 2" %>
Phylogenetic Diversity Over an Abelian Group
Andreas Dress1 and Mike Steel2
1Department for Combinatorics and Geometry, CAS-MPG Partner Institute for Computational Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences, Shanghai 200031, P.R. China
2Allan Wilson Centre for Molecular Ecology and Evolution, University of Canterbury, Christchurch, New Zealand
Annals of Combinatorics 11 (2) p.143-160 June, 2007
AMS Subject Classification: 05C05, 92D15
There is a natural way to associate to any tree T with leaf set X, and with edges weighted by elements from an abelian group G, a map from the power set of X into G―simply add the elements on the edges that connect the leaves in that subset. This map has been wellstudied in the case where G has no elements of order 2 (particularly when G is the additive group of real numbers) and, for this setting, subsets of leaves of size two play a crucial role. However, the existence and uniqueness results in that setting do not extend to arbitrary abelian groups. We study this more general problem here, and by working instead with both, pairs and triples of leaves, we obtain analogous existence and uniqueness results. Some particular results for elementary abelian 2-groups are also described.
Keywords: X-trees, split systems, abelian groups, group-valued distances, phylogenetic diversity


1. A.W.M. Dress, Split decomposition over an abelian group, Part I: Generalities, Ann. Comb., to appear.

2. H.-J. Bandelt and M.A. Steel, Symmetric matrices representable by weighted trees over a cancellative abelian monoid, SIAM J. Discrete Math. 8 (1995) 517-525.

3. G.M. Barker, Phylogenetic diversity: a quantitative framework for measurement of priority and achievement in biodiversity conservation, Biol. J. Linnean Soc. 76 (2002) 165-194.

4. S. Böcker and A.W.M. Dress, Recovering symbolically dated, rooted trees from symbolic ultrametrics, Adv. Math. 138 (1998) 105-125.

5. P. Buneman, The recovery of trees from measures of dissimilarity, In: Mathematics in the Archaeological and Historical Sciences, F.R. Hodson, D.G. Kendall, and P. Tautu, Eds., Edinburgh University Press (1971) pp. 387-395.

6. A.W.M. Dress, B. Holland, K. Huber, J. Koolen, V. Moulton, and J. Weyer-Menkhoff, D additive and D ultra-additive maps Gromov's trees, and the Farris transform, Discrete Appl. Math. 146 (2005) 51-73.

7. A.W.M. Dress, M. Hendy, K. Huber, and V. Moulton, On the number of vertices and edges of the Buneman graph, Ann. Comb. 1 (1997) 329-337.

8. A.W.M. Dress, K. Huber, and V. Moulton, Some variations on a theme by Buneman, Ann. Comb. 1 (1997) 339-352.

9. A.W.M. Dress, K. Huber, and V. Moulton, Some uses of the Farris transform in Mathematics and Phylogenetics ― A Review, Ann. Comb. 11 (2007) 1-37.

10. S.N. Evans and T.P. Speed, Invariants of some probability models used in phylogenetic inference, Ann. Statist. 21 (1993) 355-377.

11. D.P. Faith, Conservation evaluation and phylogenetic diversity, Biol. Conservation 61 (1992) 1-10.

12. J. Felsenstein, Inferring Phylogenies, Sinauer Associates Sunderland, Mass., 2004.

13. W.J. Heiser and M. Bennani, Triadic distance models: axiomatization and least squares representation, J. Math. Psych. 41 (1997) 189-206.

14. S. Joly and G. Le Calvé, Three-way distances, J. Classification 12 (1995) 191-205.

15. L. Pachter and D. Speyer, Reconstructing trees from subtree weights, Appl. Math. Lett. 17 (6) (2004) 615-621.

16. C. Semple and M.A. Steel, Tree representations of non-symmetric, group-valued proximities, Adv. Appl. Math. 23 (1999) 300-321.

17. C. Semple and M. Steel, Phylogenetics, Oxford University Press, Oxford, 2003.

18. M. Steel, Phylogenetic diversity and the greedy algorithm, Systematic Biol. 54 (2005) 527- 529.