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Mathematical models play an increasingly important role in the interpretation of biological experiments. Studies often present a model that generates the observations, connecting hypothesized process to an observed pattern. Such generative…

Tree sets are posets with additional structure that generalize tree-like objects in graphs, matroids, or other combinatorial structures. They are a special class of abstract separation systems. We study infinite tree sets and how they…

Supertree construction is the process by which a set of phylogenetic trees, each on a subset of the overall set X of species, is combined into a tree on the full set S. The traditional use of supertree methods is the assembly of a large…

A common assumption in causal modeling posits that the data is generated by a set of independent mechanisms, and algorithms should aim to recover this structure. Standard unsupervised learning, however, is often concerned with training a…

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

We develop a purely set-theoretic formalism for binary trees and binary graphs. We define a category of binary automata, and display it as a fibred category over the category of binary graphs. We also relate the notion of binary graphs to…

We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between $G$-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph $G$. A tree growing…

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…

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating…

This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting…

We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we…

We consider a model of random tree growth, where at each time unit a new vertex is added and attached to an already existing vertex chosen at random. The probability with which a vertex with degree $k$ is chosen is proportional to $w(k)$,…

Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…

Let $A$ be a commutative $k$-algebra over a field of $k$ and $\Xi$ a linear operator defined on $A$. We define a family of $A$-valued invariants $\Psi$ for finite rooted forests by a recurrent algorithm using the operator $\Xi$ and show…

Suppose $G$ is a tree. Graham's "Tree Reconstruction Conjecture" states that $G$ is uniquely determined by the integer sequence $|G|$, $|L(G)|$, $|L(L(G))|$, $|L(L(L(G)))|$, $\ldots$, where $L(H)$ denotes the line graph of the graph $H$.…

Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…

Bounded generation by root elements is a property which has been widely studied for various types of linear algebraic groups defined over rings of integers in algebraic number fields. However, when considering global function fields, there…

In this paper, we study averaging operators from an algebraic and combinatorial point of view. We first construct free averaging algebras in terms of a class of bracketed words called averaging words. We next apply this construction to…

In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner…

Sequential model-based optimization sequentially selects a candidate point by constructing a surrogate model with the history of evaluations, to solve a black-box optimization problem. Gaussian process (GP) regression is a popular choice as…