Related papers: Enumerative combinatorics of unlabeled and labeled…

Phylogenetic networks are a type of directed acyclic graph that represent how a set $X$ of present-day species are descended from a common ancestor by processes of speciation and reticulate evolution. In the absence of reticulate evolution,…

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 study a class of combinatorial objects that we call "decorated trees". These consist of vertices, arrows and edges, where each edge is decorated by two integers (one near each of its endpoints), each arrow is decorated by an integer, and…

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An…

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 consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such…

We prove that a random labeled (unlabeled) tree is balanced. We also prove that random labeled and unlabeled trees are strongly $k$-balanced for any $k\geq 3$.

Phylogenetic networks generalize phylogenetic trees by allowing the modelization of events of reticulate evolution. Among the different kinds of phylogenetic networks that have been proposed in the literature, the subclass of binary…

Inference of phylogenetic networks is of increasing interest in the genomic era. However, the extent to which phylogenetic networks are identifiable from various types of data remains poorly understood, despite its crucial role in…

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 show a first-order asymptotics result for the number of galled networks with $n$ leaves. This is the first class of phylogenetic networks of {\it large} size for which an asymptotic counting result of such strength can be obtained. In…

The ongoing explosion of genome sequence data is transforming how we reconstruct and understand the histories of biological systems. Across biological scales, from individual cells to populations and species, trees-based models provide a…

In mathematical phylogenetics, labeled histories describe the sequences by which sets of labeled lineages coalesce to a shared ancestral lineage. We study labeled histories for at-most-$r$-furcating trees. Consider a rooted leaf-labeled…

Models of growing networks are a central topic in network science. In these models, vertices are usually labeled by their arrival time, distinguishing even those node pairs whose structural roles are identical. In contrast, unlabeled…

Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of…

A normal network is uniquely determined by the set of phylogenetic trees that it displays. Given a set $\mathcal{P}$ of rooted binary phylogenetic trees, this paper presents a polynomial-time algorithm that reconstructs the unique binary…

We study the asymptotic number of certain monotonically labeled increasing trees arising from a generalized evolution process. The main difference between the presented model and the classical model of binary increasing trees is that the…

Phylogenetic networks are a generalisation of phylogenetic trees that allow for more complex evolutionary histories that include hybridisation-like processes. It is of considerable interest whether a network can be considered `tree-like' or…

In 1998, B\"{o}cker and Dress gave a 1-to-1 correspondence between symbolically dated rooted trees and symbolic ultrametrics. We consider the corresponding problem for unrooted trees. More precisely, given a tree $T$ with leaf set $X$ and a…

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…