Related papers: Protection Number in Plane Trees

We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this…

We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree…

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) of a plane partition of the positive integer n, assuming that this parfition is chosen uniformly at random from the set of all such partitions.

Uniform fluid flow distribution in a symmetric volume can be realized through a symmetric branched tree. It is shown here, however, that the flow partitioning can be highly sensitive to deviations from exact symmetry if inertial effects are…

Attack trees are a popular way to represent and evaluate potential security threats on systems or infrastructures. The goal of this work is to provide a framework allowing to express and check whether an attack tree is consistent with the…

Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. We give an efficient,…

We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to…

The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial…

Attack trees are considered a useful tool for security modelling because they support qualitative as well as quantitative analysis. The quantitative approach is based on values associated to each node in the tree, expressing, for instance,…

The aim of the current work is to prove a law of large numbers for the range size of recurrent rotor walks with random initial configuration on a general class of trees, called periodic trees or directed covers of graphs.

The degree distribution of an ordered tree $T$ with $n$ nodes is $\vec{n} = (n_0,\ldots,n_{n-1})$, where $n_i$ is the number of nodes in $T$ with $i$ children. Let $\mathcal{N}(\vec{n})$ be the number of trees with degree distribution…

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare…

We consider root-finding algorithms for random rooted trees grown by uniform attachment. Given an unlabeled copy of the tree and a target accuracy $\varepsilon > 0$, such an algorithm outputs a set of nodes that contains the root with…

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of…

We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when…

Let $T$ be a random tree taken uniformly at random from the family of labelled trees on $n$ vertices. In this note, we provide bounds for $c(n)$, the number of sub-trees of $T$ that hold asymptotically almost surely. With computer support…

Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension…

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…

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…

To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…