Noah A. Rosenberg

Models of random phylogenetic trees are important mathematical tools in phylogenetic data analysis; tree balance is a property of phylogenetic trees that helps describe and compare their shapes. Here, we investigate the Sackin index, one of the oldest and most widely used tree balance measures, under a variety of phylogenetic models beyond the standard Yule and Uniform models. The general biased speciation model is a biologically interpretable tree-generating model in which species branch into pairs of descendants according to an intrinsic branching rate, which is then split between the pair of descendants. The proportion of the branching rate of the parent that is assigned to a specific descendant, the split fraction, is a model parameter. Our main result is a formula for the expected value of the Sackin index under the general biased speciation model. Special cases of the general biased speciation model include a case that sets the split fraction to be a constant for each branching event, and two special cases inspired by the Aldous beta-splitting model: a random choice of split fraction is sampled independently at each branching event and drawn from a symmetric beta distribution in the Blum–François beta-splitting model or an asymmetric beta distribution in the Sainudiin–Véber beta-splitting model. As a corollary of our main result, we find formulas for the expected value of the Sackin index under these three cases of the general biased speciation model. We use the formulas to study how the balance of trees depends on the model parameters. Finally, because the Yule model is a special case of the Blum–François beta-splitting model, we find a new proof for the classical result describing the expected value of the Sackin index under the Yule model.

The scenario is familiar for many math students. You arrive at an organized activity among people who have not previously met, maybe a meeting of a student club, maybe a volunteering event—or any o...

Clumppling 2.0: A Clustering Alignment Program for Population Structure Analyses

The Human Genome Diversity Project (HGDP) began in 1991 as an initiative to study genetic variation from human populations worldwide. In 2002, the HGDP reported the HGDP-CEPH Human Genome Diversity Cell Line Panel, a global panel of 1064 cell lines that is maintained at the Centre d’Etude du Polymorphisme Humain (CEPH) and that has served as a major resource fundamental to the last 25 years of progress in human population genetics. HGDP-CEPH data have been central to research on topics such as human genetic diversity, human population structure, human migrations, the development of population-genetic statistics and software, and the potential value of inclusion of diverse sets of human populations in biomedical research. In this article, two researchers who participated in early analyses of genotypes from the HGDP-CEPH panel in the early 2000s speak with three researchers who played key roles in developing the Human Genome Diversity Project from its origin in 1991. The conversation reflects on the successes and challenges of the effort to launch the HGDP and on its scientific contributions.

For a specific rooted labeled tree topology, a labeled history is a sequence of branchings that give rise to that labeled topology as it unfolds over time. Here, for $r$-furcating trees, we use a connection with Huffman trees from information theory to identify maximally probable rooted trees -- unlabeled $r$-furcating topologies whose labelings each have a number of labeled histories greater than or equal to those of all other labeled topologies. Our characterization of the unique maximally probable $r$-furcating unlabeled topology generalizes the Harding--Hammersley--Grimmett result identifying the maximally probable bifurcating unlabeled topology, and it provides a new proof for that result. We present a conjecture for the maximally probable $r$-furcating unlabeled topology if labeled histories are tabulated allowing for simultaneous branching events across multiple internal nodes of a tree.