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Mathematical Hierarchies and Biology
Edited by: Boris Mirkin, Rutgers University, Piscataway, NJ, F. R. McMorris, University of Louisville, KY, Fred S. Roberts, Rutgers University, New Brunswick, NJ, and Andrey Rzhetsky, Columbia University, New York, NY
A co-publication of the AMS and DIMACS.

DIMACS: Series in Discrete Mathematics and Theoretical Computer Science
1997; 388 pp; hardcover
Volume: 37
ISBN-10: 0-8218-0762-5
ISBN-13: 978-0-8218-0762-0
List Price: US$101
Member Price: US$80.80
Order Code: DIMACS/37
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The mathematical approach to the study of hierarchies presents the theoretical basis for many important areas of current scientific investigation. Biology has benefited from this research and has also stimulated the mathematical study of hierarchies.

This collection presents papers devoted to theoretical, algorithmical, and application issues related to (1) reconstructing hierarchies (trees or ranking) from (dis)similarity or entity-to-character data, (2) using hierarchies for modeling evolution and other processes, and (3) combining (gene) trees.

The papers in this volume provide a contemporary sample of many new results in hierarchy theory with applications in biology, psychology, data analysis, and systems engineering.


  • Mathematical treatment of hierarchies in several interconnected frameworks: set systems, linear subspaces, graph objects, and tree metrics.
  • The relationship of hierarchies to many issues of current application--from learning robots to wavelets to intron evolution to the evolution of language.
  • Solutions to several important problems.

Co-published with the Center for Discrete Mathematics and Theoretical Computer Science beginning with Volume 8. Volumes 1-7 were co-published with the Association for Computer Machinery (ACM).


Graduate students and research mathematicians working in discrete mathematics, computational biology, and data analysis.

Table of Contents

  • G. F. Estabrook -- Ancestor-descendant relations and incompatible data: Motivation for research in discrete mathematics
  • C. L. Nehaniv and J. L. Rhodes -- Krohn-Rhodes theory, hierarchies & evolution
  • M. Bonet, C. Phillips, T. Warnow, and S. Yooseph -- Inferring evolutionary trees from polymorphic characters, and an analysis of the Indo-European family of languages
  • R. D. M. Page and M. A. Charleston -- Reconciled trees and incongruent gene and species trees
  • O. Eulenstein, B. Mirkin, and M. Vingron -- Comparison of annotating duplication, tree mapping, and copying as methods to compare gene trees with species trees
  • A. Rzhetsky, F. J. Ayala, L. C. Hsu, C. Chang, and A. Yoshida -- Fitting models of intron evolution to aldehyde dehydrogenase data
  • V. Moulton, M. Steel, and C. Tuffley -- Dissimilarity maps and substitution models: Some new results
  • K. Atteson -- The performance of the neighbor-joining method of phylogeny reconstruction
  • O. Gascuel -- Concerning the NJ algorithm and its unweighted version, UNJ
  • A. Guénoche -- Order distances in tree reconstruction
  • V. Makarenkov and B. Leclerc -- Circular orders of tree metrics, and their uses for the reconstruction and fitting of phylogenetic trees
  • P.-A. Landry and F.-J. Lapointe -- Estimation of missing distances in path-length matrices: Problems and solutions
  • P. M. Pardalos and X. Deng -- Complexity issues in hierarchical optimization
  • P. Hansen and D. de Werra -- Nesticity
  • F. S. Roberts and L. Sheng -- Phylogeny graphs of arbitrary digraphs
  • E. Kubicka, G. Kubicki, and F. R. McMorris -- Agreement metrics for trees revisited
  • T. M. Przytycka -- Sparse dynamic programming for maximum agreement subtree problem
  • F. R. McMorris and R. C. Powers -- The median function on weak hierarchies
  • A. W. M. Dress -- Towards a theory of holistic clustering
  • I. van Mechelen, S. Rosenberg, and P. De Boeck -- On hierarchies and hierarchical classes models
  • L. Hubert, P. Arabie, and J. Meulman -- The construction of globally optimal ordered partitions
  • J. D. Carroll and G. De Soete -- Multiple trees: Fitting two or more tree structures to proximity data
  • B. Mirkin -- Linear embedding of binary hierarchies and its applications
  • A. Meystel -- Learning algorithms generating multigranular hierarchies
  • Index
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