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We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a theorem of D\'{e}nes…
We provide a fundamental result for bucket increasing trees, which gives a complete characterization of all families of bucket increasing trees that can be generated by a tree evolution process. We also provide several equivalent…
Gene genealogies are frequently studied by measuring properties such as their height ($H$), length ($L$), sum of external branches ($E$), sum of internal branches ($I$), and mean of their two basal branches ($B$), and the coalescence times…
Statistically consistent estimation of phylogenetic trees or gene trees is possible if pairwise sequence dissimilarities can be converted to a set of distances that are proportional to the true evolutionary distances. Susko et al. (2004)…
We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex…
A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called…
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 evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation…
We investigate the spectral properties of rooted trees with the intention of improving the currently existing results that deal with this matter. The concept of an assigned rational function is recursively defined for each vertex of a…
A wired tree is a graph obtained from a tree by collapsing the leaves to a single vertex. We describe a pair of short exact sequences relating the sandpile group of a wired tree to the sandpile groups of its principal subtrees. In the case…
In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees.
Regression trees have emerged as a preeminent tool for solving real-world regression problems due to their ability to deal with nonlinearities, interaction effects and sharp discontinuities. In this article, we rather study regression trees…
A proper vertex of a rooted tree with totally ordered vertices is a vertex that is less than all its proper descendants. We count several kinds of labeled rooted trees and forests by the number of proper vertices. Our results are all…
The note contains a short elementary proof of Cayley's formula for labeled trees.
Based on decision trees, many fields have arguably made tremendous progress in recent years. In simple words, decision trees use the strategy of "divide-and-conquer" to divide the complex problem on the dependency between input features and…
This article studies sums over all compositions of an integer. We derive a generating function for this quantity, and apply it to several special functions, including various generalized Bernoulli numbers. We connect composition sums with a…
In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear…
We explore from an algebraic viewpoint the properties of the tree languages definable with a first-order formula involving the ancestor predicate, using the description of these languages as those recognized by iterated block products of…
This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree…
Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of \textit{ordered Reeb graphs}, every Reeb graph with $n$ leaves and first…