We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour.where B(a, b) = Γ(a)Γ(b)/Γ(a + b) is the Beta function and Z n , n ≥ 2, are normalisation constants; this extends to β = −2 by continuity, i.e. q Aldous

In this paper, we present a new way to describe the timing of branching events in phylogenetic trees. Our description is in terms of the relative timing of diversification events between sister clades; as such it is complementary to existing methods using lineages-through-time plots which consider diversification in aggregate. The method can be applied to look for evidence of diversification happening in lineage-specific "bursts", or the opposite, where diversification between 2 clades happens in an unusually regular fashion. In order to be able to distinguish interesting events from stochasticity, we discuss 2 classes of neutral models on trees with relative timing information and develop a statistical framework for testing these models. These model classes include both the coalescent with ancestral population size variation and global rate speciation-extinction models. We end the paper with 2 example applications: first, we show that the evolution of the hepatitis C virus deviates from the coalescent with arbitrary population size. Second, we analyze a large tree of ants, demonstrating that a period of elevated diversification rates does not appear to have occurred in a bursting manner.

This paper deals with several bijections between cladograms and perfect matchings. The first of these is due to Diaconis and Holmes. The second is a modification of the Diaconis-Holmes matching which makes deletion of the largest labeled leaf correspond to gluing together the last two points in the perfect matching. The third is an entirely new encoding of cladograms, first as a bijection with a certain set of strings and then via this to perfect matchings. In this pair of bijections, deletion of the largest labeled leaf corresponds to deletion of the corresponding symbols from the string or deletion of the corresponding pair from the matching. These two new bijections are related through a common max-min labeling of internal vertices with two different choices for the label of the root node. All these encodings are extended to cladograms with edge lengths and left-right ordered children. Moving a single symbol in this last encoding corresponds to a subtree prune and regraft operation on the cladogram, making it well suited for use in phylogentics software. Finally, a perfect Gray code for cladograms is derived from the bar encoding, along with a total ordering on all cladograms, Algorithms are also provided for finding the next and previous cladogram, the cladogram at any position, and the position of any cladogram in the sequence.

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The reconstruction of large phylogenetic trees from data that violates clocklike evolution (or as a supertree constructed from any m input trees) raises a difficult question for biologists–how can one assign relative dates to the vertices of the tree? In this paper we investigate this problem, assuming a uniform distribution on the order of the inner vertices of the tree (which includes, but is more general than, the popular Yule distribution on trees). We derive fast algorithms for computing the probability that (i) any given vertex in the tree was the j–th speciation event (for each j), and (ii) any one given vertex is earlier in the tree than a second given vertex. We show how the first algorithm can be used to calculate the expected length of any given interior edge in any given tree that has been generated under either a constant-rate speciation model, or the coalescent model.