Related papers: Mass-structure of weighted real trees
Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles,…
A simple and computationally efficient scheme for tree-structured vector quantization is presented. Unlike previous methods, its quantization error depends only on the intrinsic dimension of the data distribution, rather than the apparent…
Binary trees are fundamental objects in models of evolutionary biology and population genetics. Here, we discuss some of their combinatorial and structural properties as they depend on the tree class considered. Furthermore, the process by…
We assess advantages of expressing tree-structured Ising models via their mean parameterization rather than their commonly chosen canonical parameterization. This includes fixedness of marginal distributions, often convenient for dependence…
Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…
In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various…
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…
Periodic trees are combinatorial structures which are in bijection with cluster tilting objects in cluster categories of affine type $\tilde{A}_{n-1}$. The internal edges of the tree encode the $c$-vectors corresponding to the cluster…
Hierarchical tree structures are common in many real-world systems, from tree roots and branches to neuronal dendrites and biologically inspired artificial neural networks, as well as in technological networks for organizing and searching…
We show that for many models of random trees, the independence number divided by the size converges almost surely to a constant as the size grows to infinity; the trees that we consider include random recursive trees, binary and $m$-ary…
The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix. The class of regular matroids generalizes that of graphical matroids, and a…
In order to develop statistical methods for shapes with a tree-structure, we construct a shape space framework for tree-like shapes and study metrics on the shape space. This shape space has singularities, corresponding to topological…
We prime-encode the natural numbers via recursive factorisation, iterated to the exponents, generating a corpus of planar rooted trees equivalently represented as Dyck words. This forms a deterministic text endowed with internal rules.…
The well-known Lawvere category R of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But R has another such structure, given by multiplication, which is *-autonomous. Normed sets,…
This paper studies the "explanation problem" for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A whose…
We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in…
Let $T$ be an infinite rooted tree with weights $w_e$ assigned to its edges. Denote by $m_n(T)$ the minimum weight of a path from the root to a node of the $n$th generation. We consider the possible behaviour of $m_n(T)$ with focus on the…
The Ward numbers $W(n,k)$ combinatorially enumerate set partitions with block sizes $\geq 2$ and phylogenetic trees (total partition trees). We prove that $W(n,k)$ also counts \emph{increasing Schr\"oder trees} by verifying they satisfy…
In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and define a new notion of equivalence of discrete Morse…
The Random Projection Tree structures proposed in [Freund-Dasgupta STOC08] are space partitioning data structures that automatically adapt to various notions of intrinsic dimensionality of data. We prove new results for both the RPTreeMax…