Related papers: Asymptotic height of Plancherel random trees

Given a metric space on n points, an {\alpha}-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of…

Let $K(=K_{n,\theta})$ be a positive integer-valued random variable whose distribution is given by ${\rm P}(K = x) = \bar{s}(n,x) \theta^x/(\theta)_n$ $(x=1,\ldots,n) $, where $\theta$ is a positive number, $n$ is a positive integer,…

This paper investigates the computational geometry relevant to calculations of the Frechet mean and variance for probability distributions on the phylogenetic tree space of Billera, Holmes and Vogtmann, using the theory of probability…

A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special…

A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We study the behavior of measurable…

It is well-known that the largest eigenvalue of an $n\times n$ GUE matrix and the length of a longest increasing subsequence in a uniform random permutation of length $n$, both converge weakly to the GUE Tracy-Widom distribution as $n\to…

In this paper, we investigate random walks in a family of small-world trees having an exponential degree distribution. First, we address a trapping problem, that is, a particular case of random walks with an immobile trap located at the…

Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n tends to infinty. The same is true for the number of vertices of depth divisible by m for m=3, 4 or 5; in all…

We study an asymptotical behavior of the maximal degree in the degree distribution in an evolving tree model combining the local choice and the Mori's preferential attachment. In the considered model, the random graph is constructed in the…

We prove asymptotic normality for the number of fringe subtrees isomorphic to any given tree in uniformly random trees with given vertex degrees. As applications, we also prove corresponding results for random labelled trees with given…

We investigate properties of node centrality in random growing tree models. We focus on a measure of centrality that computes the maximum subtree size of the tree rooted at each node, with the most central node being the tree centroid. For…

Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the…

This article focuses on properties and structures of trees with maximum mean subtree order in a given family; such trees are called optimal in the family. Our main goal is to describe the structure of optimal trees in $\mathcal{T}_n$ and…

If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with…

We prove that if T is a tree on n vertices wih maximum degree D and the edge probability p(n) satisfies: np>c*max{D*logn,n^{\epsilon}} for some constant \epsilon>0, then with high probability the random graph G(n,p) contains a copy of T.…

Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under…

We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted P\'olya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic…

In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative $(1+\delta)$ of uniform in expected…

For every sufficiently well-behaved function $g:\mathbb{R}_{\ge 0}\rightarrow\mathbb{R}_{\ge 0}$ that grows at least linearly and at most exponentially we construct a tree $T$ of uniform volume growth $g$, that is, $$C_1\cdot g(r/4)\le…

Let $\mathcal{P}_k(n)$ denote the set of partitions of $n$ whose largest part is bounded by $k,$ which are in well-known bijection with $(k+1)$-cores $\mathcal{C}_k$. We study a growth process on $\mathcal{C}_k$, whose stationary…