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We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and…
Survival analysis studies and predicts the time of death, or other singular unrepeated events, based on historical data, while the true time of death for some instances is unknown. Survival trees enable the discovery of complex nonlinear…
An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of bins are related to the split times of a Yule process. The asymptotic behavior of the…
Measures of tree balance play an important role in various research areas, for example in phylogenetics. There they are for instance used to test whether an observed phylogenetic tree differs significantly from a tree generated by the Yule…
User-generated family trees are invaluable for constructing population-scale family networks and studying population dynamics over many generations and far into the past. Family trees contain information on individuals such as birth and…
The Robinson-Foulds (RF) metric is arguably the most widely used measure of phylogenetic tree similarity, despite its well-known shortcomings: For example, moving a single taxon in a tree can result in a tree that has maximum distance to…
The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade)…
We present new links between some remarkable martingales found in the study of the Binary Search Tree, or of the Bisection Problem, looking at them on the probability space of a continuous time binary branching process.
Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of…
For a particular case of a branching random walk with lattice support, namely the Yule branching random walk, we prove that the distribution of the centred maximum oscillates around a distribution corresponding to a critical travelling wave…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict…
In this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution…
Phylogenetic networks are an extension of phylogenetic trees that allow for the representation of reticulate evolution events. One of the classes of networks that has gained the attention of the scientific community over the last years is…
Statistically consistent estimation of phylogenetic trees or gene trees is possible if pairwise sequence dissimilarities can be converted to a set of distances that are proportional to the true evolutionary distances. Susko et al. (2004)…
Galled trees are studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into tree-child, galled and reticulation-visible network classes by relaxing a structural condition imposed on…
With the aim of considering models with persistent memory we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macrovolution. Here the model is analyzed and interpreted in the framework…
A learning algorithm is presented which given the structure of a causal tree, will estimate its link probabilities by sequential measurements on the leaves only. Internal nodes of the tree represent conceptual (hidden) variables…
We study the distribution of the number of leaves of the subtree chosen uniformly at random among all the subtrees of the critical branching process tree at extinction.
We investigate how the pattern of contacts between species in mutualistic ecosystems is affected by the phylogenetic proximity between the species of each guild. We develop a dynamical model geared to establish the role of such proximity in…