A functional limit theorem for the profile of bb-ary trees

In this paper we prove a functional limit theorem for the weighted profile of a $b$-ary tree. For the proof we use classical martingales connected to branching Markov processes and a generalized version of the profile-polynomial martingale. By embedding, choosing weights and a branch factor in a right way, we finally rediscover the profiles of some well-known discrete time trees.Comment: Published in at http://dx.doi.org/10.1214/09-AAP640 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

End-use related physical and mechanical properties of selected fast-growing poplar hybrids (Populus trichocarpa x P-deltoides)

This study focused on physical and mechanical properties of fast-growing poplar clones in relation to potential end uses with high added value. A total of 14 trees from three different clones, all P. trichocarpa x deltoides (T x D) hybrids, were felled in a poplar plantation in Lille (Belgium): six 'Beaupre', four 'Hazendans' and four 'Hoogvorst'. Growth rate was found to have no significant influence on the physical mechanical properties. Although the investigated clones are genetically closely related, important variations in physical and mechanical properties were observed. Specific features such as spatial distribution of tension wood and dimensional stability are the main quality factors. It was concluded that 'Beaupre' is suitable for a wide range of high value added applications, such as plywood or construction wood. 'Hazendans' and 'Hoogvorst' will need adapted technology in processing. Further research is needed to characterize clonally induced variation in properties and to assess adequate processing strategies for multiclonal poplar stands

Long and short paths in uniform random recursive dags

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

Minima in branching random walks

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results fully characterize the possible behavior of $\mathbf {E}M_n$ when the branching random walk has bounded branching and step size.Comment: Published in at http://dx.doi.org/10.1214/08-AOP428 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Network analysis of a corpus of undeciphered Indus civilization inscriptions indicates syntactic organization

Archaeological excavations in the sites of the Indus Valley civilization (2500-1900 BCE) in Pakistan and northwestern India have unearthed a large number of artifacts with inscriptions made up of hundreds of distinct signs. To date there is no generally accepted decipherment of these sign sequences and there have been suggestions that the signs could be non-linguistic. Here we apply complex network analysis techniques to a database of available Indus inscriptions, with the aim of detecting patterns indicative of syntactic organization. Our results show the presence of patterns, e.g., recursive structures in the segmentation trees of the sequences, that suggest the existence of a grammar underlying these inscriptions.Comment: 17 pages (includes 4 page appendix containing Indus sign list), 14 figure

On weighted depths in random binary search trees

Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyze weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search process, the weighted path length and the weighted Wiener index in a random binary search tree. We establish three regimes of nodes depending on whether the second order behaviour of their weighted depths follows from fluctuations of the keys on the path, the depth of the nodes, or both. Finally, we investigate a random distribution function on the unit interval arising as scaling limit for weighted depths of nodes with at most one child

Spatial Smoothing Techniques for the Assessment of Habitat Suitability

Precise knowledge about factors influencing the habitat suitability of a certain species forms the basis for the implementation of effective programs to conserve biological diversity. Such knowledge is frequently gathered from studies relating abundance data to a set of influential variables in a regression setup. In particular, generalised linear models are used to analyse binary presence/absence data or counts of a certain species at locations within an observation area. However, one of the key assumptions of generalised linear models, the independence of the observations is often violated in practice since the points at which the observations are collected are spatially aligned. While several approaches have been developed to analyse and account for spatial correlation in regression models with normally distributed responses, far less work has been done in the context of generalised linear models. In this paper, we describe a general framework for semiparametric spatial generalised linear models that allows for the routine analysis of non-normal spatially aligned regression data. The approach is utilised for the analysis of a data set of synthetic bird species in beech forests, revealing that ignorance of spatial dependence actually may lead to false conclusions in a number of situations

Asymptotics of heights in random trees constructed by aggregation

To each sequence $(a_n)$ of positive real numbers we associate a growing sequence $(T_n)$ of continuous trees built recursively by gluing at step $n$ a segment of length $a_n$ on a uniform point of the pre-existing tree, starting from a segment $T_1$ of length $a_1$. Previous works on that model focus on the influence of $(a_n)$ on the compactness and Hausdorff dimension of the limiting tree. Here we consider the cases where the sequence $(a_n)$ is regularly varying with a non-negative index, so that the sequence $(T_n)$ exploses. We determine the asymptotics of the height of $T_n$ and of the subtrees of $T_n$ spanned by the root and $\ell$ points picked uniformly at random and independently in $T_n$, for all $\ell \in \mathbb N$