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This study introduces a novel recursive method for efficiently generating binary orchard networks and tree-child networks. This method simplifies network generation by reducing it to the creation of unique integer sequences, aiding evolutionary studies.

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Area of Science:

  • Computational Biology
  • Evolutionary Biology
  • Discrete Mathematics

Background:

  • Phylogenetic trees model evolutionary history but cannot represent reticulate evolution.
  • Phylogenetic networks extend trees to include reticulate events like hybridization.
  • Orchard networks generalize tree-child networks, a well-studied class of phylogenetic networks.

Purpose of the Study:

  • To address the combinatorial and algorithmic challenge of generating binary orchard and tree-child networks.
  • To develop an efficient method for enumerating these networks based on their structural properties.

Main Methods:

  • Networks are characterized by reversing a reduction process, leading to a unique representation.
  • This unique representation is based on sequences of integer pairs, related to network size (leaves and reticulations).
  • A recursive algorithm is developed for generating all minimal such sequences.

Main Results:

  • The proposed recursive method efficiently generates all minimal integer sequences.
  • This directly enables the generation of all binary orchard and tree-child networks for a given number of leaves and reticulations.
  • The number of binary orchard networks with up to 6 leaves and 8 reticulations was computed using an implemented algorithm.

Conclusions:

  • The developed method provides an efficient way to generate and enumerate phylogenetic orchard and tree-child networks.
  • This facilitates further research into the combinatorial properties and evolutionary implications of these network structures.
  • The availability of an open-source implementation aids practical application and computational experiments in phylogenetics.