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Tree-ordered weakly sparse models have recently emerged as a robust framework for representing structures in an ``almost sparse'' way, while allowing the structure to be reconstructed through a simple first-order interpretation. A prominent…
In this work we study the interleaving distance between merge trees from a combinatorial point of view. We use a particular type of matching between trees to obtain a novel formulation of the distance. With such formulation, we tackle the…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…
In this work, we introduce a random field in view of natural image modeling, obtained as a limit of sequences of dead leaves models, when considering arbitrarily small or big objects. The dead leaves model, introduced by the Mathematical…
Dual-tree algorithms are a widely used class of branch-and-bound algorithms. Unfortunately, developing dual-tree algorithms for use with different trees and problems is often complex and burdensome. We introduce a four-part logical split:…
We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with…
In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is…
A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n…
Let F be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then F must contain uncountably many non-compact leaves. We prove the same statement for oriented p-dimensional foliations of…
Words in some natural languages can have a composite structure. Elements of this structure include the root (that could also be composite), prefixes and suffixes with which various nuances and relations to other words can be expressed.…
A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety.…
We discuss a new notion of pattern avoidance motivated by the operad theory: pattern avoidance in planar labelled trees. It is a generalisation of various types of consecutive pattern avoidance studied before: consecutive patterns in words,…
This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without…
The degree distribution of an ordered tree $T$ with $n$ nodes is $\vec{n} = (n_0,\ldots,n_{n-1})$, where $n_i$ is the number of nodes in $T$ with $i$ children. Let $\mathcal{N}(\vec{n})$ be the number of trees with degree distribution…
Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices,…
Many websites with an underlying database containing structured data provide the richest and most dense source of information relevant for topical data integration. The real data integration requires sustainable and reliable pattern…
In this paper, we estimate the weak saturation numbers of trees. As a case study, we examine caterpillars and obtain several tight estimates. In particular, this implies that for any $\alpha\in [1,2]$, there exist caterpillars with $k$…
We propose a new topological invariant of unlabeled trees of N nodes. The invariant is a set of Nx2 matrices of integers, with sum_j k^{d_{i,j}} and v_i as the matrix elements, where d_{i,j} are the elements of the distance matrix and v_i…