Related papers: The Proportion of Trees that are Linear

Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges,…

We define fully non-perturbative generalizations of the uniform density and comoving curvature perturbations, which are known, in the linear theory, to be conserved on sufficiently large scales for adiabatic perturbations. Our non-linear…

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is…

A uniform recursive tree on $n$ vertices is a random tree where each possible $(n-1)!$ labeled recursive rooted tree is selected with equal probability. In this paper we introduce and study weighted trees, a non-uniform recursive tree model…

We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…

We consider a class of density-dependent branching processes which generalises exponential, logistic and Gompertz growth. A population begins with a single individual, grows exponentially initially, and then growth may slow down as the…

We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this…

Plane increasing trees are rooted labeled trees embedded into the plane such that the sequence of labels is increasing on any branch starting at the root. Relaxed binary trees are a subclass of unlabeled directed acyclic graphs. We…

We show that there exists an outerplanar graph on $O(n^{c})$ vertices for $c = \log_2(3+\sqrt{10}) \approx 2.623$ that contains every tree on $n$ vertices as a subgraph. This extends a result of Chung and Graham from 1983 who showed that…

A closed-form formula is derived for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also…

We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and…

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…

To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargava's factorials to a natural combinatorial setting suitable for studying questions around…

In a recent paper, McDiarmid, Semple, and Welsh (2015) showed that the number of tree-child networks with $n$ leaves has the factor $n^{2n}$ in its main asymptotic growth term. In this paper, we improve this by completely identifying the…

Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…

We investigate the relation between degree sequences of trees and the majorization order using the Muirhead theorem. In this way, we prove a theorem that provides a necessary and sufficient condition for delta sequences of trees to be…

Rooted binary perfect phylogenies provide a generalization of rooted binary unlabeled trees in which each leaf is assigned a positive integer value that corresponds in a biological setting to the count of the number of indistinguishable…

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre-existing tree, starting from a…

We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using…

For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem…