Related papers: Classifying Tree Topologies along Tropical Line Se…
Classification of gene trees is an important task both in the analysis of multi-locus phylogenetic data, and assessment of the convergence of Markov Chain Monte Carlo (MCMC) analyses used in Bayesian phylogenetic tree reconstruction. The…
Galled trees are studied as a recombination model in theoretic population genetics. This class of phylogenetic networks has been generalized to tree-child networks, normal networks and tree-based networks by relaxing a structural condition.…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI)…
Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice…
This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each…
Phylogenomics is a new field which applies to tools in phylogenetics to genome data. Due to a new technology and increasing amount of data, we face new challenges to analyze them over a space of phylogenetic trees. Because a space of…
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…
Tropical geometry and the theory of Newton-Okounkov bodies are two methods which produce toric degenerations of an irreducible complex projective variety. Kaveh-Manon showed that the two are related. We give geometric maps between the…
In order to conduct a statistical analysis on a given set of phylogenetic gene trees, we often use a distance measure between two trees. In a statistical distance-based method to analyze discordance between gene trees, it is a key to decide…
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to…
We investigate the tree gonality of a genus-$g$ metric graph, defined as the minimum degree of a tropical morphism from any tropical modification of the metric graph to a metric tree. We give a combinatorial constructive proof that this…
Many popular algorithms for searching the space of leaf-labelled trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are…
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…
The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning. Established comparison methods treat the standard setting that the distributions are supported in the same space. Recently,…
Previous work indicates that tropical forest can exist as an alternative stable state to savanna. Therefore, perturbation by climate change or human impact may lead to crossing of a tipping point beyond which there is rapid forest dieback…
A bijection between ternary trees with $n$ nodes and a subclass of Motzkin paths of length $3n$ is given. This bijection can then be generalized to $t$-ary trees.
In this work, we introduce a Tropical Axial Attention neural reasoning architecture that replaces vanilla softmax dot-product attention with max-plus operators, inducing a piecewise-linear structure aligned with dynamic programming…
An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of…