Abstract
Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number $a_n(\sigma)$ of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with $n\geq 1$ leaves, fixed (for the relabelling action) by a given permutation $\sigma\in\frak{S}_n$. Denoting by $\lambda\vdash n$ the integer partition giving the sizes of the cycles of $\sigma$ in non-increasing order, they show by a guessing/checking approach that if $\lambda$ is a binary partition (it is known that $a_n(\sigma)=0$ otherwise), then$$a_n(\sigma)=\prod_{i=2}^{\ell(\lambda)}(2(\lambda_i+\cdots+\lambda_{\ell(\lambda)})-1),$$and they derive from it a formula and random generation procedure for tanglegrams (and more generally for tangled chains). Our main result is a combinatorial proof of the formula for $a_n(\sigma)$, which yields a simplification of the random sampler for tangled chains.
Highlights
For A a finite set of cardinality n ≥ 1, we denote by B[A] the set of rooted binary trees that are non-embedded and have n leaves with distinct labels from A
Such trees are known as phylogenetic trees, where typically A is the set of represented species
The isomorphism (2) can be seen as an adaptation of Remy’s method [7] to the setting of binary trees fixed by a given permutation
Summary
For A a finite set of cardinality n ≥ 1, we denote by B[A] the set of rooted binary trees that are non-embedded (i.e., the order of the two children of each node does not matter) and have n leaves with distinct labels from A. Such trees are known as phylogenetic trees, where typically A is the set of represented species. The isomorphism (2) can be seen as an adaptation of Remy’s method [7] to the setting of (non-embedded rooted) binary trees fixed by a given permutation.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.