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The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes…

Phylogenetic networks are used to represent the evolutionary history of species. Recently, the new class of orchard networks was introduced, which were later shown to be interpretable as trees with additional horizontal arcs. This makes the…

The reliability of a phylogenetic inference method from genomic sequence data is ensured by its statistical consistency. Bayesian inference methods produce a sample of phylogenetic trees from the posterior distribution given sequence data.…

The presence of reticulate evolutionary events in phylogenies turn phylogenetic trees into phylogenetic networks. These events imply in particular that there may exist multiple evolutionary paths from a non-extant species to an extant one,…

Compatibility of phylogenetic trees is the most important concept underlying widely-used methods for assessing the agreement of different phylogenetic trees with overlapping taxa and combining them into common supertrees to reveal the tree…

This paper presents a detailed comparison of a recently proposed algorithm for optimizing decision trees, tree alternating optimization (TAO), with other popular, established algorithms. We compare their performance on a number of…

This work briefly explores the possibility of approximating spatial distance (alternatively, similarity) between data points using the Isolation Forest method envisioned for outlier detection. The logic is similar to that of isolation: the…

The {\em edit distance} between two ordered trees with vertex labels is the minimum cost of transforming one tree into the other by a sequence of elementary operations consisting of deleting and relabeling existing nodes, as well as…

There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…

Edit distances between merge trees of scalar fields have many applications in scientific visualization, such as ensemble analysis, feature tracking or symmetry detection. In this paper, we propose branch mappings, a novel approach to the…

This paper introduces a novel stability measure for edit distances between merge trees of piecewise linear scalar fields. We apply the new measure to various metrics introduced recently in the field of scalar field comparison in scientific…

Given two binary trees on $N$ labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartets distance between the two trees is…

This paper addresses the problem of finding a representation of a subtree distance, which is an extension of the tree metric. We show that a minimal representation is uniquely determined by a given subtree distance, and give a linear time…

Scalar field comparison is a fundamental task in scientific visualization. In topological data analysis, we compare topological descriptors of scalar fields -- such as persistence diagrams and merge trees -- because they provide succinct…

Hybridization networks are representations of evolutionary histories that allow for the inclusion of reticulate events like recombinations, hybridizations, or lateral gene transfers. The recent growth in the number of hybridization network…

Estimating phylogenetic trees is an important problem in evolutionary biology, environmental policy and medicine. Although trees are estimated, their uncertainties are discarded by mathematicians working in tree space. Here we explicitly…

The tree-metric theorem provides a necessary and sufficient condition for a dissimilarity matrix to be a tree metric, and has served as the foundation for numerous distance-based reconstruction methods in phylogenetics. Our main result is…

Determining the interaction partners among protein/domain families poses hard computational problems, in particular in the presence of paralogous proteins. Available approaches aim to identify interaction partners among protein/domain…

We develop combinatorial methods for computing the rotation distance between binary trees, i.e., equivalently, the flip distance between triangulations of a polygon. As an application, we prove that, for each n, there exist size n trees at…

Tree structures appear in many fields of the life sciences, including phylogenetics, developmental biology and nucleic acid structures. Trees can be used to represent RNA secondary structures, which directly relate to the function of…