Related papers: Counting trees using symmetries
The reconstruction of a central tendency `species tree' from a large number of conflicting gene trees is a central problem in systematic biology. Moreover, it becomes particularly problematic when taxon coverage is patchy, so that not all…
Phylogenetic trees are a central tool in understanding evolution. They are typically inferred from sequence data, and capture evolutionary relationships through time. It is essential to be able to compare trees from different data sources…
This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case…
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general…
We propose a statistical method to test whether two phylogenetic trees with given alignments are significantly incongruent. Our method compares the two distributions of phylogenetic trees given by the input alignments, instead of comparing…
A cluster tree provides a highly-interpretable summary of a density function by representing the hierarchy of its high-density clusters. It is estimated using the empirical tree, which is the cluster tree constructed from a density…
The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history…
Topological phylogenetic trees can be assigned edge weights in several natural ways, highlighting different aspects of the tree. Here the rooted triple and quartet metrizations are introduced, and applied to formulate novel fast methods of…
Staged trees are a recently-developed, powerful family of probabilistic graphical models. An equivalence class of staged trees has now been characterised, and two fundamental statistical operators have been defined to traverse the…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…
We present convincing empirical evidence for an effective and general strategy for building accurate small models. Such models are attractive for interpretability and also find use in resource-constrained environments. The strategy is to…
Inspired by [4] we present a new algorithm for uniformly random generation of ordered trees in which all occuring outdegrees can be specified by a given sequence of numbers. The method can be used for random generation of binary or n-ary…
The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…
Data structures known as $k$-d trees have numerous applications in scientific computing, particularly in areas of modern statistics and data science such as range search in decision trees, clustering, nearest neighbors search, local…
We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in…
Recent work has proven the existence of extreme inbreeding in a European ancestry sample taken from the contemporary UK population \cite{nature_01}. This result brings our attention again to a math problem related to inbreeding family trees…
Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical…
We develop optimal algorithms for learning undirected Gaussian trees and directed Gaussian polytrees from data. We consider both problems of distribution learning (i.e. in KL distance) and structure learning (i.e. exact recovery). The first…
A metric phylogenetic tree relating a collection of taxa induces weighted rooted triples and weighted quartets for all subsets of three and four taxa, respectively. New intertaxon distances are defined that can be calculated from these…
P\'olya trees are rooted, unlabeled trees on $n$ vertices. This paper gives an efficient, new way to generate P\'olya trees. This allows comparing typical unlabeled and labeled tree statistics and comparing asymptotic theorems with…