Related papers: Galton-Watson Trees with the Same Mean Have the Sa…
A set $P\subset \mathbb N$ is called predictive if for any zero entropy finite-valued stationary process $(X_i)_{i\in \mathbb Z}$, $X_0$ is measurable with respect to $(X_i)_{i\in P}$. We know that $\mathbb N$ is a predictive set. In this…
Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $\lambda$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the…
The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$,…
Under minimal condition, we prove the local convergence of a critical multi-type Galton-Watson tree conditioned on having a large total progeny by types towards a multi-type Kesten's tree. We obtain the result by generalizing Neveu's strong…
In mathematical phylogenetics, the time-consistent galled trees provide a simple class of rooted binary network structures that can be used to represent a variety of different biological phenomena. We study the enumerative combinatorics of…
The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. In this…
We study the long-term behavior of weighted multi-type branching processes, focusing on extending classical laws of large numbers and martingale convergence to settings with infinitely many weighted particles, arbitrary type spaces and…
In this article, we introduce a formal definition of the concept of probability tree and conduct a detailed and comprehensive study of its fundamental structural properties. In particular, we define what we term an inductive probability…
We show that if a strictly positive joint probability distribution for a set of binary random variables factors according to a tree, then vertex separation represents all and only the independence relations enclosed in the distribution. The…
A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\meg 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. A vector homogeneous tree $\mathbf{T}$…
We consider conditioned Galton-Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe…
We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic…
We consider the Ising model on a supercritical Galton-Watson tree $\mathbf{T}_n$ of depth $n$ with a sparse random external field, given by a collection of i.i.d. Bernouilli random variables with vanishing parameter $p_n$. This may me…
We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the…
The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$. We prove under general assumptions…
The study of Gaussian free field level sets on supercritical Galton-Watson trees has been initiated by Ab\"acherli and Sznitman in Ann. Inst. Henri Poincar\'{e} Probab. Stat., 54(1):173--201, 2018. By means of entirely different tools, we…
We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton-Watson tree conditioned to be large…
We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…
Conformal prediction is a statistically rigorous method for quantifying uncertainty in models by having them output sets of predictions, with larger sets indicating more uncertainty. However, prediction sets are not inherently actionable;…
Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical…