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Title: Enumeration of Rooted Binary Unlabeled Galled Trees
Abstract

Rooted binarygalledtrees generalize rooted binary trees to allow a restricted class of cycles, known asgalls. We build upon the Wedderburn-Etherington enumeration of rooted binary unlabeled trees withnleaves to enumerate rooted binary unlabeled galled trees withnleaves, also enumerating rooted binary unlabeled galled trees withnleaves andggalls,$$0 \leqslant g \leqslant \lfloor \frac{n-1}{2} \rfloor $$0gn-12. The enumerations rely on a recursive decomposition that considers subtrees descended from the nodes of a gall, adopting a restriction on galls that amounts to considering only the rooted binarynormalunlabeled galled trees in our enumeration. We write an implicit expression for the generating function encoding the numbers of trees for alln. We show that the number of rooted binary unlabeled galled trees grows with$$0.0779(4.8230^n)n^{-\frac{3}{2}}$$0.0779(4.8230n)n-32, exceeding the growth$$0.3188(2.4833^n)n^{-\frac{3}{2}}$$0.3188(2.4833n)n-32of the number of rooted binary unlabeled trees without galls. However, the growth of the number of galled trees with only one gall has the same exponential order 2.4833 as the number with no galls, exceeding it only in the subexponential term,$$0.3910n^{\frac{1}{2}}$$0.3910n12compared to$$0.3188n^{-\frac{3}{2}}$$0.3188n-32. For a fixed number of leavesn, the number of gallsgthat produces the largest number of rooted binary unlabeled galled trees lies intermediate between the minimum of$$g=0$$g=0and the maximum of$$g=\lfloor \frac{n-1}{2} \rfloor $$g=n-12. We discuss implications in mathematical phylogenetics.

 
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NSF-PAR ID:
10496570
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Bulletin of Mathematical Biology
Volume:
86
Issue:
5
ISSN:
0092-8240
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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