Related papers: A Cantor-Bendixson Rank for Siblings of Trees

We study probability distributions over free algebras of trees. Probability distributions can be seen as particular (formal power) tree series [Berstel et al 82, Esik et al 03], i.e. mappings from trees to a semiring K . A widely studied…

By weighted tree we understand such connected tree,that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we…

We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…

In 1998, B\"{o}cker and Dress gave a 1-to-1 correspondence between symbolically dated rooted trees and symbolic ultrametrics. We consider the corresponding problem for unrooted trees. More precisely, given a tree $T$ with leaf set $X$ and a…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

This paper is concerned with the approximation of high-dimensional functions in a statistical learning setting, by empirical risk minimization over model classes of functions in tree-based tensor format. These are particular classes of…

We prove the Tree Alternative Conjecture for the topological minor relation: letting $[T]$ denote the equivalence class of $T$ under the topological minor relation we show that: $|[T]| = 1$ or $|[T]|\geq \aleph_0$ and $\forall r\in V(T)$,…

This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree…

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary…

A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…

Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille…

We examine a discrete random recursive tree growth process that, at each time step, either adds or deletes a node from the tree with probability $p$ and $1-p$, respectively. Node addition follows the usual uniform attachment model. For node…

The Bel-Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov-Bel classification of the Weyl tensor. The new classes…

Modeling of multidimensional signal using tensor is more convincing than representing it as a collection of matrices. The tensor based approaches can explore the abundant spatial and temporal structures of the mutlidimensional signal. The…

We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we…

A central limit theorem for binary tree is numerically examined. Two types of central limit theorem for higher-order branches are formulated. A topological structure of a binary tree is expressed by a binary sequence, and the…

We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $\mathbb{H}$ in the $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. We describe a decomposition of $T$ into $3$ simple tensors in the $2\times 2\times…

Generalized trees, we call them O-trees, are defined as hierarchical partial orders, i.e., such that the elements larger than any one are linearly ordered. Quasi-trees are, roughly speaking, undirected O-trees. For O-trees and quasi-trees,…

This paper contains a classification of countable lower 1-transitive linear orders. The notion of lower 1-transitivity generalises that of 1-transitivity for linear orders, and is essential for the structure theory of 1-transitive trees.…

We give a representation for labeled ordered trees that supports labeled queries such as finding the i-th ancestor of a node with a given label. Our representation is succinct, namely the redundancy is small-o of the optimal space for…