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A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in…

Efforts to reconstruct phylogenetic trees and understand evolutionary processes depend fundamentally on stochastic models of speciation and mutation. The simplest continuous-time model for speciation in phylogenetic trees is the Yule…

In this paper, we use the H2 norm associated with a communication graph to characterize the robustness of consensus to noise. In particular, we restrict our attention to trees and by systematic attention to the effect of local changes in…

We extend some results of Itai Benjamini and Yuri Lima (see \href{http://arxiv.org/pdf/1305.2610.pdf}{\cite{Benjamini}}). In this paper they consider a binary tree $\mathbb T_n$ of height $n$, each leaf is either infected by one of $k$…

In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch…

Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place $k$ arcs between the edges of a rooted binary phylogenetic…

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree $T$ on $n$ leaves with a path metric $d$, consider the pairwise distances $\{d(u,v)\}$ between leaves. It is well known that these…

Null models of binary phylogenetic trees are useful for testing hypotheses on real world phylogenies. In this paper we consider phylogenies as binary trees without edge lengths together with a sampling measure and encode them as algebraic…

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled…

A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random…

Let $G$ be a simple strongly connected weighted directed graph. Let $\mathcal{G}$ denote the spanning tree graph of $G$. That is, the vertices of $\mathcal{G}$ consist of the directed rooted spanning trees on $G$, and the edges of…

We investigate random connected graphs from a block-stable class whose distribution is weighted based on the number of $2$-connected components, or blocks. This includes the class of planar graphs. For this, we develop a notion of a…

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the…

We study a family of tree-type diagrams that arise in studies of the cumulant expansion in discrete Erd\H os-R\'enyi random matrix models. Using a version of the Pr\" ufer code, we obtain an explicit expression for the number of tree-type…

In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic…

Let T d (n) be the set of d-ary rooted trees with n internal nodes. We give a method to construct a sequence (t n , n $\ge$ 0) where, for any n $\ge$ 1, t n has the uniform distribution in T d (n), and t n is constructed from t n--1 by the…

Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set {1,...,n}. Define the k-weights of the tree D_{i_1,....,i_k}(T) as the sum of the lengths of the edges of the minimal subtree connecting i_1,....,i_k. We…

We study the convergence of the predictive surface of regression trees and forests. To support our analysis we introduce a notion of adaptive concentration for regression trees. This approach breaks tree training into a model selection…

We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the…

We investigate multivariate regular variation in the context of time-homogeneous Markov chains on general vector spaces and in random coefficient linear models. In the first part, we show that the regular variation of the stationary…