Enumeration of rooted binary perfect phylogenies

Abstract

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees in which each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable lineages associated with the leaf. For the rooted binary unlabeled trees, these integers equal 1. We address a variety of enumerative problems concerning rooted binary perfect phylogenies with sample size $s$: the rooted binary unlabeled trees in which a sample of size $s$ lineages is distributed across the leaves of an unlabeled tree with $n$ leaves, $1 \leq n \leq s$. The enumerations further characterize the rooted binary perfect phylogenies, which include the rooted binary unlabeled trees, and which can provide a set of structures useful for various biological contexts.

Figures (5)

• Figure 1: A perfect phylogeny with sample size $s=17$ and $n=8$ leaves. The numbers at the leaves represent leaf multiplicities.
• Figure 2: The lattice of rooted binary perfect phylogenies for sample size $s=5$. Each column is labeled by its associated number of leaves $n$.
• Figure 3: The number $b_s$ of perfect phylogenies with sample size $s$. Exact values are computed from eq. \ref{['eq:enumeration_alln']}. The asymptotic approximation is computed from eq. \ref{['eq:approx']}.
• Figure 4: The enumeration of all $b_{8,6}=61$ rooted binary perfect phylogenies with sample size $s=8$ and $n=6$ leaves. The number of leaves in the right subtree is indicated by $j$, and $i$ indicates the sample size for the right subtree.
• Figure 5: The $b_{s,3}$ rooted binary perfect phylogenies with $n=3$ leaves, for each $s$ from 3 to 8, as obtained by Proposition \ref{['prop:b_s,n recursion']}. The value of $i$ indicates the sample size for the right subtree.

Theorems & Definitions (27)

• proposition 1
• proposition 2
• lemma 3
• proof
• corollary 4
• proof
• theorem 5
• proof
• proposition 6
• proof