Related papers: Correction terms for the height of weighted recurs…
In the past several years a wide range of methods for the construction of regression trees and other estimators based on the recursive partitioning of samples have appeared in the statistics literature. Many applications involve data…
For each integer $k \geq 2$, we introduce a sequence of $k$-ary discrete trees constructed recursively by choosing at each step an edge uniformly among the present edges and grafting on "its middle" $k-1$ new edges. When $k=2$, this…
We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case…
This thesis examines linearly edge-reinforced random walks on infinite trees. In particular, recurrence and transience of such random walks on general (fixed) trees as well as on Galton-Watson trees (i.e. random trees) is characterized, and…
We first consider the growth of trees by probabilistic attachment of new vertices to leaves. This leads to a growth model based on vertex clusters and probabilities assigned to clusters. This model turns out to be readily applicable to…
Random spanning trees are among the most prominent determinantal point processes. We give four examples of random spanning trees on ladder-like graphs whose rungs form stationary renewal processes or regenerative processes of order two,…
A compacted binary tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special…
We establish the large deviation probabilities for the height of random recursive trees, revealing polynomial upper-tail decay and stretched-exponential lower-tail decay. Remarkably, the lower tail features an atypical prefactor that grows…
In this paper, we study a class of random walks that build their own tree. At each step, the walker attaches a random number of leaves to its current position. The model can be seen as a subclass of the Random Walk in Changing Environments…
Given $n \in \mathbb{N}$ and $\mu \in \mathbb{R}$, a $\textit{$\mu$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-\mu h(t)}$, where…
In the hyperbolic plane there are infinite regular lattices. From a fix vertex of a lattice tree graphs can be constructed recursively to the next layers with edges of the lattice. In this article we examine the properties of the growing of…
Topological phylogenetic trees can be assigned edge weights in several natural ways, highlighting different aspects of the tree. Here the rooted triple and quartet metrizations are introduced, and applied to formulate novel fast methods of…
Inspired by scientific collaboration networks, especially our empirical analysis of the network of econophysicists, an evolutionary model for weighted networks is proposed. Both degree-driven and weight-driven models are considered.…
Dynamic regression trees are an attractive option for automatic regression and classification with complicated response surfaces in on-line application settings. We create a sequential tree model whose state changes in time with the…
We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples…
Given an ensemble of randomized regression trees, it is possible to restructure them as a collection of multilayered neural networks with particular connection weights. Following this principle, we reformulate the random forest method of…
Traditional studies of memory for meaningful narratives focus on specific stories and their semantic structures but do not address common quantitative features of recall across different narratives. We introduce a statistical ensemble of…
Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical…
This paper studies the problem of estimating the order of arrival of the vertices in a random recursive tree. Specifically, we study two fundamental models: the uniform attachment model and the linear preferential attachment model. We…
We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree…