Related papers: Asymptotic results on the length of coalescent tre…

We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-\alpha,\alpha)$-coalescent tree for $1<\alpha<2$, starting from $n$ individuals. There are two regimes: If $\alpha\le1/2(1+\sqrt{5})$, then $L_n$…

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy…

In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce as they are traced back in time. As the…

We investigate the statistics of trees grown from some initial tree by attaching links to preexisting vertices, with attachment probabilities depending only on the valence of these vertices. We consider the asymptotic mass distribution that…

For $1<\alpha <2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-\alpha, \alpha)$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch…

We derive the asymptotic behavior of the total, active and inactive branch lengths of the seed bank coalescent, when the size of the initial sample grows to infinity. Those random variables have important applications for populations…

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of…

We consider a ${\Lambda}$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. ${\Lambda}={\delta}_0$, the result is well…

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths),…

We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of…

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…

We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is…

This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as Xi-coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number…

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic…

A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the…

We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each…

We consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give…

Coalescent histories provide lists of species tree branches on which gene tree coalescences can take place, and their enumerative properties assist in understanding the computational complexity of calculations central in the study of gene…

We study the asymptotic distribution of integers sharing the same rooted-tree structure that encodes their complete prime factorization tower. For each tree we derive an explicit density formula depending only on a pair $(m,k)$, the density…

Phylogenetic trees are widely used to understand the evolutionary history of organisms. Tree shapes provide information about macroevolutionary processes. However, macroevolutionary models are unreliable for inferring the true processes…