Related papers: Random expansions of trees with bounded height

We begin with a new analysis of formal words. Let w be a formal word in letters g_1,...,g_k. The word map associated with w maps the permutations s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every occurrence of…

This study is dedicated to precise distributional analyses of the height of non-plane unlabelled binary trees ("Otter trees"), when trees of a given size are taken with equal likelihood. The height of a rooted tree of size $n$ is proved to…

Let's denote a complete $m$-ary rooted tree graph of height $n$ as $G$. In scope of this paper we prove the certain relations between the properties of $G$ and the expectation and variance of the distribution of lengths of strings,…

The height of a random PATRICIA tree built from independent, identically distributed infinite binary strings with arbitrary diffuse probability distribution $\mu$ on $\{0,1\}^\mathbb{N}$ is studied. We show that the expected height grows…

Given a class of graphs F, we say that a graph G is universal for F, or F-universal, if every H in F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of…

We prove that for any fixed $k$, the probability that a random vertex of a random increasing plane tree is of rank $k$, that is, the probability that a random vertex is at distance $k$ from the leaves, converges to a constant $c_k$ as the…

Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total…

Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $\epsilon$ and $\Delta$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree…

A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…

We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [Berstel et al 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widely studied…

We consider a (random permutation model) binary search tree with n nodes and give asymptotics on the loglog scale for the height H_n and saturation level h_n of the tree as n\to\infty, both almost surely and in probability. We then consider…

We consider a regular $n$-ary tree of height $h$, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we…

Consider a family of random ordered graph trees $(T_n)_{n\geq 1}$, where $T_n$ has $n$ vertices. It has previously been established that if the associated search-depth processes converge to the normalised Brownian excursion when rescaled…

We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result…

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…

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…

Let $T_\lambda$ be a Galton--Watson tree with Poisson($\lambda$) offspring, and let $A$ be a tree property. In this paper, are concerned with the regularity of the function $\mathbb{P}_\lambda(A):= \mathbb{P}(T_\lambda \vdash A)$. We show…

In the critical beta-splitting model of a random $n$-leaf rooted tree, clades are recursively split into sub-clades, and a clade of $m$ leaves is split into sub-clades containing $i$ and $m-i$ leaves with probabilities $\propto 1/(i(m-i))$.…

The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous…

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage $n$ is a non-negative integer. For any given vertex $x$ of SG(n), we derive rigorously the probability distribution of the degree $j \in \{1,2,3,4\}$ at the…