Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula to count th
e number of distinct binary rooted tanglegrams with $n$ matched vertices, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This includes a new formula for the number of binary trees with $n$ leaves. We also give a conjecture for the expected number of cherries in a large randomly chosen binary tree and an extension of this conjecture to other types of trees.
Many discrete mathematics problems in phylogenetics are defined in terms of the relative labeling of pairs of leaf-labeled trees. These relative labelings are naturally formalized as tanglegrams, which have previously been an object of study in coevo
lutionary analysis. Although there has been considerable work on planar drawings of tanglegrams, they have not been fully explored as combinatorial objects until recently. In this paper, we describe how many discrete mathematical questions on trees factor through a problem on tanglegrams, and how understanding that factoring can simplify analysis. Depending on the problem, it may be useful to consider a unordered version of tanglegrams, and/or their unrooted counterparts. For all of these definitions, we show how the isomorphism types of tanglegrams can be understood in terms of double cosets of the symmetric group, and we investigate their automorphisms. Understanding tanglegrams better will isolate the distinct problems on leaf-labeled pairs of trees and reveal natural symmetries of spaces associated with such problems.
The human microbiome is the ensemble of genes in the microbes that live inside and on the surface of humans. Because microbial sequencing information is now much easier to come by than phenotypic information, there has been an explosion of sequencing
and genetic analysis of microbiome samples. Much of the analytical work for these sequences involves phylogenetics, at least indirectly, but methodology has developed in a somewhat different direction than for other applications of phylogenetics. In this paper I review the field and its methods from the perspective of a phylogeneticist, as well as describing current challenges for phylogenetics coming from this type of work.
In microbial ecology studies, the most commonly used ways of investigating alpha (within-sample) diversity are either to apply count-only measures such as Simpsons index to Operational Taxonomic Unit (OTU) groupings, or to use classical phylogenetic
diversity (PD), which is not abundance-weighted. Although alpha diversity measures that use abundance information in a phylogenetic framework do exist, but are not widely used within the microbial ecology community. The performance of abundance-weighted phylogenetic diversity measures compared to classical discrete measures has not been explored, and the behavior of these measures under rarefaction (sub-sampling) is not yet clear. In this paper we compare the ability of various alpha diversity measures to distinguish between different community states in the human microbiome for three different data sets. We also present and compare a novel one-parameter family of alpha diversity measures, BWPD_theta, that interpolates between classical phylogenetic diversity (PD) and an abundance-weighted extension of PD. Additionally, we examine the sensitivity of these phylogenetic diversity measures to sampling, via computational experiments and by deriving a closed form solution for the expectation of phylogenetic quadratic entropy under re-sampling. In all three of the datasets considered, an abundance-weighted measure is the best differentiator between community states. OTU-based measures, on the other hand, are less effective in distinguishing community types. In addition, abundance-weighted phylogenetic diversity measures are less sensitive to differing sampling intensity than their unweighted counterparts. Based on these results we encourage the use of abundance-weighted phylogenetic diversity measures, especially for cases such as microbial ecology where species delimitation is difficult.
