Related papers: Enumeration of multi-rooted plane trees

In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in…

We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows…

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.

We introduce some natural families of distributions on rooted binary ranked plane trees with a view toward unifying ideas from various fields, including macroevolution, epidemiology, computational group theory, search algorithms and other…

Given an integer $M\geq 2$, we deploy the generating function techniques to compute the number of $M$-th roots of identity in some of the well-known finite groups of Lie type, more precisely for finite general linear groups, symplectic…

We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible…

We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.

We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions.…

Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…

Phylogenetic trees are binary nonplanar trees with labelled leaves, and plane oriented recursive trees are planar trees with an increasing labelling. Both families are enumerated by double factorials. A bijection is constructed, using the…

We obtain first order linear partial differential equations which are satisfied by exponential generating functions of two variables for the number of connected bipartite graphs with given Betti number. By solving these equations…

Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…

We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…

We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation,…

Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests.…

The grammar representation of a narrowing tree for a syntactically deterministic conditional term rewriting system and a pair of terms is a regular tree grammar that generates expressions for substitutions obtained by all possible…

To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function \Phi_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described…

We consider natural polynomial truncations of hypergeometric power series defined over finite fields. For these truncations, we establish asymptotic upper bounds of order $O(p^{11/12})$ on the number of roots in the prime field…

Let $p(m)$ (respectively, $q(m)$) be the maximum number $k$ such that any tree with $m$ edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with $k$ edges. We derive closed-form expressions…