Related papers: Uniform, rigid branchwise-real trees
In this paper we study planar morphs between straight-line planar grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
This paper studies the structure of graphs with given tree-width and excluding a fixed complete bipartite subgraph, which generalises the bounded degree setting. We give a new structural description of such graphs in terms of so-called…
In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.
We propose a new approach to image segmentation, which exploits the advantages of both conditional random fields (CRFs) and decision trees. In the literature, the potential functions of CRFs are mostly defined as a linear combination of…
If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly…
We develop a theoretical framework for the analysis of oblique decision trees, where the splits at each decision node occur at linear combinations of the covariates (as opposed to conventional tree constructions that force axis-aligned…
To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the…
We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The…
The Brownian continuum tree was extensively studied in the 90s as a universal random metric space. One construction obtains the continuum tree by a change of metric from an excursion function (or continuous circle mapping) on $[0,1]$. This…
Given a graph $G$, the Bell $k$-coloring graph $\mathcal{B}_k(G)$ has vertices given by partitions of $V(G)$ into $k$ independent sets (allowing empty parts), with two partitions adjacent if they differ only in the placement of a single…
The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform…
We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph…
We study notions of hyperuniformity for invariant locally square-integrable point processes in regular trees. We show that such point processes are never geometrically hyperuniform, and if the diffraction measure has support in the…
Motivated by online recommendation systems, we study a family of random forests. The vertices of the forest are labeled by integers. Each non-positive integer $i\le 0$ is the root of a tree. Vertices labeled by positive integers $n \ge 1$…
We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and…
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…
Consider a random real tree whose leaf set, or boundary, is endowed with a finite mass measure. Each element of the tree is further given a type, or allele, inherited from the most recent atom of a random point measure…
An evolutionary tree is a cascade of bifurcations starting from a single common root, generating a growing set of daughter species as time goes by. Species here is a general denomination for biological species, spoken languages or any other…
Let $G$ be a simple finite connected graph with vertex set $V(G) = \{v_1,v_2,\ldots,v_n\}$. Denote the degree of vertex $v_i$ by $d_i$ for all $1 \leq i \leq n$. The Randi\'c matrix of $G$, denoted by $R(G) = [r_{i,j}]$, is the $n \times n$…