Related papers: Tree statistics from Matula numbers
We count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.
In this work a composition-decomposition technique is presented that correlates tree eigenvectors with certain eigenvectors of an associated so-called skeleton forest. In particular, the matching properties of a skeleton determine the…
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…
A random forest is a popular tool for estimating probabilities in machine learning classification tasks. However, the means by which this is accomplished is unprincipled: one simply counts the fraction of trees in a forest that vote for a…
Rooted phylogenetic networks are often constructed by combining trees, clusters, triplets or characters into a single network that in some well-defined sense simultaneously represents them all. We review these four models and investigate…
Binary rooted trees, both in the ordered and in the un-ordered case, are well studied structures in the field of combinatorics. The aim of this work is to study particular patterns in these classes of trees. We consider completely…
Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. But, as far as…
Consensus methods provide a useful strategy for combining information from a collection of gene trees. An important application of consensus methods is to combine gene trees to estimate a species tree. To investigate the theoretical…
The Horton-Strahler number -- also called the register function -- is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton-Strahler number of stable Galton-Watson trees conditioned…
A hierarchical structure describing the inter-relationships of species has long been a fundamental concept in systematic biology, from Linnean classification through to the more recent quest for a 'Tree of Life.' In this paper we use an…
For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem…
We study multicolour, oriented and high-dimensional discrepancies of the set of all subtrees of a tree. As our main result, we show that the $r$-colour discrepancy of the subtrees of any tree is a linear function of the number of leaves…
Networks having the geometry and the connectivity of trees are considered as the spatial support of spatiotemporal dynamical processes. A tree is characterized by two parameters: its ramification and its depth. The local dynamics at the…
We introduce the set of (non-spanning) tree-decorated planar maps, and show that they are in bijection with the Cartesian product between the set of trees and the set of maps with a simple boundary. As a consequence, we count the number of…
A phylogenetic tree shows the evolutionary relationships among species. Internal nodes of the tree represent speciation events and leaf nodes correspond to species. A goal of phylogenetics is to combine such trees into larger trees, called…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
For random matrices with tree-like structure there exists a recursive relation for the local Green functions whose solution permits to find directly many important quantities in the limit of infinite matrix dimensions. The purpose of this…
For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges…
A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random…
In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this…