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Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

  • Olivier Bodini
  • Antoine GenitriniEmail author
  • Mehdi NaimaEmail author
  • Alexandros Singh
Conference paper
  • 88 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12159)

Abstract

There exists a wealth of literature concerning families of increasing trees, particularly suitable for representing the evolution of either data structures in computer science, or probabilistic urns in mathematics, but are also adapted to model evolutionary trees in biology. The classical notion of increasing trees corresponds to labeled trees such that, along paths from the root to any leaf, node labels are strictly increasing; in addition nodes have distinct labels. In this paper we introduce new families of increasingly labeled trees relaxing the constraint of unicity of each label. Such models are especially useful to characterize processes evolving in discrete time whose nodes evolve simultaneously. In particular, we obtain growth processes for biology much more adequate than the previous increasing models. The families of monotonic trees we introduce are much more delicate to deal with, since they are not decomposable in the sense of Analytic Combinatorics. New tools are required to study the quantitative statistics of such families. In this paper, we first present a way to combinatorially specify such families through evolution processes, then, we study the tree enumerations.

Keywords

Analytic Combinatorics Asymptotic enumeration Increasing trees Monotonic trees Borel transform Evolution process 

Notes

Acknowledgment

We thank Stephan Wagner for a fruitful discussion about the relationship of an involved proof of this paper and the article [8]. Furthermore we are grateful for the anonymous reviewers whose comments and suggestions helped improving and clarifying this manuscript.

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Copyright information

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Authors and Affiliations

  1. 1.Université Sorbonne Paris Nord, Laboratoire d’Informatique de Paris Nord, CNRS, UMR 7030VilletaneuseFrance
  2. 2.Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6 -LIP6- UMR 7606ParisFrance

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