Related papers: A note on hook length formulas for trees
Motivated by the properties of the descent polynomials, which enumerate permutations of $S_n$ with a fixed descent set, we define descent polynomials for labeled rooted trees. We give recursive and explicit formulas for these polynomials…
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number…
We give characterizations for the parabolicity of regular trees.
The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform…
We give a precise description of combed trees in terms of Kelly-Mac Lane graphs. We show that any combed tree is uniquely expressed as an allowable Kelly-Mac Lane graph of a certain shape. Conversely, we show that any such Kelly-Mac Lane…
We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives…
There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…
The article describes the structural and algorithmic relations between Cartesian trees and Lyndon Trees. This leads to a uniform presentation of the Lyndon table of a word corresponding to the Next Nearest Smaller table of a sequence of…
Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This paper surveys several graph parameters tied to treewidth, including separation number, tangle number, well-linked number and Cartesian…
We consider so-called simple families of labelled trees, which contain, e.g., ordered, unordered, binary and cyclic labelled trees as special instances, and study the global and local behaviour of the number of inversions. In particular we…
The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of a regular graph to…
We study the complexity and expressive power of conjunctive queries over unranked labeled trees represented using a variety of structure relations such as ``child'', ``descendant'', and ``following'' as well as unary relations for node…
We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher…
In a recent paper on 'Estimating Species Trees from Unrooted Gene Trees' Liu and Yu observe that the distance matrix on the underlying taxon set, which is built up from expected internode distances on gene trees under the multispecies…
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 provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an…
This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…