Generating functions for multi-labeled trees

Authors:
í. Czabarka;P. L. Erds;V. Johnson;V. Moulton
Affiliations:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States;Rényi Institute of Mathematics, Budapest, Hungary;Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States;School of Computing Sciences, University of East Anglia, Norwich, NR4 7TJ, UK
Venue:
Discrete Applied Mathematics
Year:
2013

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Abstract

Multi-labeled trees are a generalization of phylogenetic trees that are used, for example, in the study of gene versus species evolution and as the basis for phylogenetic network construction. Unlike phylogenetic trees, in a leaf-multi-labeled tree it is possible to label more than one leaf by the same element of the underlying label set. In this paper we derive formulae for generating functions of leaf-multi-labeled trees and use these to derive recursions for counting such trees. In particular, we prove results which generalize previous theorems by Harding on so-called tree-shapes, and by Otter on relating the number of rooted and unrooted phylogenetic trees.