Related papers: Counting trees using symmetries
This paper is concerned with the reliable inference of optimal tree-approximations to the dependency structure of an unknown distribution generating data. The traditional approach to the problem measures the dependency strength between…
We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a…
Galled trees are widely studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into galled networks by relaxing a structural condition. In this work, a linear recurrence formula is given…
A caterpillar tree is a connected, acyclic, graph in which all vertices are either a member of a central path, or joined to that central path by a single edge. In other words, caterpillar trees are the class of trees which become path…
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…
We introduce the notion of quota trees in directed graphs. Given a nonnegative integer ``quota'' for each vertex of a directed multigraph $G$, a quota tree is an immersed rooted tree which hits each vertex of $G$ the prescribed number of…
One can often make inferences about a growing network from its current state alone. For example, it is generally possible to determine how a network changed over time or pick among plausible mechanisms explaining its growth. In practice,…
Eliciting preferences from human judgements is inherently imprecise, yet most decision analysis methods force a single priority vector from pairwise comparisons, discarding the information embedded in inconsistencies. We instead leverage…
This article presents a new way to understand the descriptive ability of tree shape statistics. Where before tree shape statistics were chosen by their ability to distinguish between macroevolutionary models, the ``resolution'' presented in…
We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings…
We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give…
This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linear-Gaussian models,…
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A…
Tree-based ensemble methods such as random forests, gradient-boosted trees, and Bayesianadditive regression trees have been successfully used for regression problems in many applicationsand research studies. In this paper, we study ensemble…
Combinatoric formulas for cluster expansions have been improved many times over the years. Here we develop some new combinatoric proofs and extensions of the tree formulas of Brydges and Kennedy, and test them on a series of pedagogical…
Spanning trees are relevant to various aspects of networks. Generally, the number of spanning trees in a network can be obtained by computing a related determinant of the Laplacian matrix of the network. However, for a large generic…
Counting the number of spanning trees in specific classes of graphs has attracted increasing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained in the 2020s. The main method is…
This paper derives a unifying theorem establishing consistency results for a broad class of tree-based algorithms. It improves current results in two aspects. First of all, it can be applied to algorithms that vary from traditional Random…
As researchers collect increasingly large molecular data sets to reconstruct the Tree of Life, the heterogeneity of signals in the genomes of diverse organisms poses challenges for traditional phylogenetic analysis. A class of phylogenetic…
In this work, we present a series of bijections that reveal the deep connections between the concepts of tree records, the girth of a connected endofunction, and the genesis sequence, the first sequence in the OEIS. We use these results to…