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A contemporary and fundamental problem faced by many evolutionary biologists is how to puzzle together a collection $\mathcal P$ of partial trees (leaf-labelled trees whose leaves are bijectively labelled by species or, more generally,…
Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone so-called reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network…
The ability to compare complex systems can provide new insight into the fundamental nature of the processes captured in ways that are otherwise inaccessible to observation. Here, we introduce the $n$-tangle method to directly compare two…
Fitch graphs $G=(X,E)$ are digraphs that are explained by $\{\emptyset, 1\}$-edge-labeled rooted trees $T$ with leaf set $X$: there is an arc $(x,y) \in E$ if and only if the unique path in $T$ that connects the last common ancestor…
Random forests on the one hand, and neural networks on the other hand, have met great success in the machine learning community for their predictive performance. Combinations of both have been proposed in the literature, notably leading to…
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…
Orthology and paralogy relations are often inferred by methods based on gene similarity, which usually yield a graph depicting the relationships between gene pairs. Such relation graphs are known to frequently contain errors, as they cannot…
Phylogenetic trees represent certain species and their likely ancestors. In such a tree, present-day species are leaves and an edge from u to v indicates that u is an ancestor of v. Weights on these edges indicate the phylogenetic distance.…
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…
In this paper we introduce a variation on the multidimensional segment tree, formed by unifying different interpretations of the dimensionalities of the data structure. We give some new definitions to previously well-defined concepts that…
Connected acyclic graphs (trees) are data objects that hierarchically organize categories. Collections of trees arise in a diverse variety of fields, including evolutionary biology, public health, machine learning, social sciences and…
We present approximation algorithms for the following NP-hard optimization problems related to bottleneck spanning trees in metric spaces. 1. The disjoint bottleneck spanning tree problem: Given $n$ pairs of points in a metric space, find…
This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of…
Neural networks with tree-based sentence encoders have shown better results on many downstream tasks. Most of existing tree-based encoders adopt syntactic parsing trees as the explicit structure prior. To study the effectiveness of…
In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set…
We consider determinantal point processes (DPPs) constrained by spanning trees. Given a graph $G=(V,E)$ and a positive semi-definite matrix $\mathbf{A}$ indexed by $E$, a spanning-tree DPP defines a distribution such that we draw…
Deep neural networks have proved to be a very effective way to perform classification tasks. They excel when the input data is high dimensional, the relationship between the input and the output is complicated, and the number of labeled…
Unrooted phylogenetic networks are graphs used to represent evolutionary relationships. Accurately reconstructing such networks is of great relevance for evolutionary biology. It has recently been conjectured that all phylogenetic networks…
We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a…
This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph…