Plane Trees Research Articles

Representing elements of the Weyl algebra by labeled trees

It is shown how arbitrary elements of the Weyl algebra can be represented by labeled plane trees using normal ordering. Several examples are treated and combinatorial aspects are discussed. Also, possible avenues for future research are described.

Pattern Popularity in 132-Avoiding Permutations

The popularity of a pattern $p$ is the total number of copies of $p$ within all permutations of a set. We address popularity in the set of $132$-avoidng permutations. Bóna showed that in this set, all other non-monotone length-$3$ patterns are equipopular, and proved equipopularity relations between some length-$k$ patterns of a specific form. We prove equipopularity relations between general length-$k$ patterns, based on the structure of their corresponding binary plane trees. Our result explains all equipopularity relations for patterns of length up to $7$, and we conjecture that it provides a complete classification of equipopularity in $132$-avoiding permutations.

Evaluation of entomopathogenic fungi against the sycamore lace bug, Corythucha ciliata (Say) (Hemiptera: Tingidae)

The sycamore lace bug, Corythucha ciliata (Say) (Hemiptera: Tingidae), is one of the most destructive pests of plane trees (Platanus spp.) worldwide. This pest is a major nuisance in Europe since plane trees are very popular in parks. Thirteen entomopathogenic fungal strains including 4 isolates of Beauveria bassiana, 2 isolates of Beauveria pseudobassiana, 6 isolates of Metarhizium anisopliae, and 1 isolate of Isaria fumosorosea were screened against adults and nymphs of C. ciliata under controlled laboratory conditions in order to test their efficacy and search for an effective and safe biocontrol agent strategy. Each isolate was applied with a conidial concentration of 1 × 107 conidia mL-1 to adults and nymphs of the pest. B. bassiana isolate KTU - 24 showed the highest mortality for both adults and nymphs with 86% within 2 weeks after inoculation. This isolate also caused the highest mycosis for adults and nymphs with 83% and 80%, respectively. Mortalities of the other fungal isolates ranged from 43% to 86% and from 36% to 73% for adults and nymphs. Therefore, B. bassiana KTU - 24 was selected for further dose mortality tests based on its high virulence and mycosis value. Dose-response mortality bioassays using 5 different concentrations (1 × 104, 1 × 105, 1 × 106, 1 × 107, and 1 × 108 conidia mL-1) were performed to determine the lethal concentration (LC50) of this isolate. Based on probit analysis, the LC50 values of isolate B. bassiana KTU - 24 were calculated as 5.51 × 105 and 3.96 × 105 conidia mL-1 against adults and nymphs, respectively. Consequently, B. bassiana KTU - 24 appears to be a promising candidate for further investigations as a biocontrol agent against C. ciliata.

Biogeochemistry of Plane Trees as a Tool to Detect Atmospheric Pollution

The plane tree, which is a valuable tool to detect atmospheric pollution, is one of the most common trees in European cities. Soil and leaf samplings were carried out in Barcelona and its environs (NE Spain) to establish the soil-plant relationship. Dry and ashed leaves and soils were analyzed by Instrumental Neutron Activation Analysis and Inductively Coupled Plasma-Optical Emission Spectroscopy (ICP/OES) at the ACTLABS laboratories in Ontario, Canada. Given that diesel is the main fuel used in vehicles in Europe, we sought to establish the role of diesel in atmospheric pollution. Diesel samples were obtained from service stations and analyzed after preconcentration using ICP/MS at the geochemistry laboratories of the University of Barcelona. The average content of diesel oil shows high values of Pb, Cu, Cr, Ag, Cd and Mn. High values of Pb, Cu, Au, Hg and Sb in leaves and soils were detected downtown and along main roads outside the city, whereas low levels of these elements were observed in rural areas.

A combinatorial invariant for escape time Sierpiński rational maps

An escape time Sierpinski map is a rational map drawn from the McMullen family z 7→ zn + λ/zn with escaping critical orbits and Julia set homeomorphic to the Sierpinski curve continuum. We address the problem of characterizing postcritically finite escape time Sierpinski maps in a combinatorial way. To accomplish this, we define a combinatorial model given by a planar tree whose vertices come with a pair of combinatorial data that encodes the dynamics of critical orbits. We show that each escape time Sierpinski map realizes a subgraph of the combinatorial tree and the combinatorial information is a complete conjugacy invariant. Dedicated to Joyce T. Macabea, in loving memory.

A CONTOUR TREE BASED VISUALIZATION FOR EXPLORING DATA WITH UNCERTAINTY

Uncertainty is a common and crucial issue in scientific data. The exploration and analysis of three-dimensional (3D) and large two-dimensional (2D) data with uncertainty information demand an effective visualization augmented with both user interaction and relevant context. The contour tree has been exploited as an efficient data structure to guide exploratory visualization. This paper proposes an interactive visualization tool for exploring data with quantitative uncertainty representations. First, we introduce a balanced planar hierarchical contour tree layout integrated with tree view interaction, allowing users to quickly navigate between levels of detail for contours of large data. Further, uncertainty information is attached to a planar contour tree layout to avoid the visual cluttering and occlusion in viewing uncertainty in 3D data or large 2D data. For the first time, the uncertainty information is explored as a combination of the data-level uncertainty which represents the uncertainty concerning the numerical values of the data, the contour variability which quantifies the positional variation of contours, and the topology variability which reveals the topological variation of contour trees. This information provides a new insight into how the uncertainty exists with and relates to the features of the data. The experimental results show that this new visualization facilitates a quick and accurate selection of prominent contours with high or low uncertainty and variability.

Divisible Transition Systems and Multiplanar Dependency Parsing

Transition-based parsing is a widely used approach for dependency parsing that combines high efficiency with expressive feature models. Many different transition systems have been proposed, often formalized in slightly different frameworks. In this article, we show that a large number of the known systems for projective dependency parsing can be viewed as variants of the same stack-based system with a small set of elementary transitions that can be composed into complex transitions and restricted in different ways. We call these systems divisible transition systems and prove a number of theoretical results about their expressivity and complexity. In particular, we characterize an important subclass called efficient divisible transition systems that parse planar dependency graphs in linear time. We go on to show, first, how this system can be restricted to capture exactly the set of planar dependency trees and, secondly, how the system can be generalized to k-planar trees by making use of multiple stacks. Using the first known efficient test for k-planarity, we investigate the coverage of k-planar trees in available dependency treebanks and find a very good fit for 2-planar trees. We end with an experimental evaluation showing that our 2-planar parser gives significant improvements in parsing accuracy over the corresponding 1-planar and projective parsers for data sets with non-projective dependency trees and performs on a par with the widely used arc-eager pseudo-projective parser.

Adsorption of Methylene Blue by NaOH-modified Dead Leaves of Plane Trees

NaOH-modified dead leaves of plane trees were used as bioadsorbent to remove methylene blue (MB) from aqueous solution. Variable influencing factors, including contact time, temperature, initial MB concentration and pH were studied through single-factor experiments. The results showed that the initial concentration 100 mg/L, bioadsorbent of 2.5 g/L, pH of 7, room temperature were the best adsorption conditions. The NaOH-modified bioadsorbent had a high adsorption capacity for MB, and its saturated extent of adsorption was 203.28 mg/g, which was higher than the un-modified dead leaves (145.62 mg/g) and some other bioadsorbents. Finally, adsorption kinetics and isotherms were discussed, suggesting that the Langmuir isotherm model and Pseudo-second order kinetics were fitted well with the adsorption process.

Connected sum at infinity and Cantrell-Stallings hyperplane unknotting

We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary proof of a remarkable theorem of J. C. Cantrell and J. R. Stallings. It asserts unknotting of proper embeddings of euclidean (m-1)-space in euclidean m-space with m not equal to 3, for all three classical manifold categories: topological, piecewise linear, and differentiable. It is one of the few major theorems whose statement and proof can be the same for all three categories. We give it the acronym HLT, which is short for Hyperplane Linearization Theorem. The topological version of the HLT immediately implies B. Mazur's topological Schoenflies theorem. We can thus claim that the Cantrell-Stallings theorem, as we present it, is an enhancement of the topological Schoenflies theorem that has exceptional didactic value. We also prove a classification of multiple codimension 1 hyperplane embeddings in eucldiean m-space for m not equal to 3. Namely, they are classified by countable simplicial trees with one edge for each hyperplane (planar trees for m=2). This result is called the Multiple Hyperplane Linearization Theorem, or MHLT for short. We give an exposition of C. Greathouse's Slab Theorem, and in conclusion some possibly novel proofs of the 2-dimensional MHLT and related results classifying contractible 2-manifolds with boundary.

Asymptotics of trees with a prescribed degree sequence and applications

AbstractLettbe a rooted tree andnbi(t) the number of nodes inthavingichildren. The degree sequenceoftsatisfies, wheredenotes the number of nodes int. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence; we writefor the corresponding distribution. Letbe a list of degree sequences indexed by κ corresponding to trees with size. We show that under some simple and natural hypotheses onthe trees sampled underconverge to the Brownian continuum random tree after normalisation by. Some applications concerning Galton–Watson trees and coalescence processes are provided.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 290‐316, 2014

HEURISTICAL FEATURE EXTRACTION FROM LIDAR DATA AND THEIR VISUALIZATION

Abstract. Extraction of landscape features from LiDAR data has been studied widely in the past few years. These feature extraction methodologies have been focussed on certain types of features only, namely the bare earth model, buildings principally containing planar roofs, trees and roads. In this paper, we present a methodology to process LiDAR data through DBSCAN, a density based clustering method, which extracts natural and man-made clusters. We then develop heuristics to process these clusters and simplify them to be sent to a visualization engine.

Tail bounds for the height and width of a random tree with a given degree sequence

AbstractFix a sequence c = (c1,…,cn) of non‐negative integers with sum n − 1. We say a rooted tree T has child sequence c if it is possible to order the nodes of T as v1,…,vn so that for each 1 ≤ i ≤ n, vi has exactly ci children. Let \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}${\mathcal T}$\end{document} be a plane tree drawn uniformly at random from among all plane trees with child sequence c. In this note we prove sub‐Gaussian tail bounds on the height (greatest depth of any node) and width (greatest number of nodes at any single depth) of \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}${\mathcal T}$\end{document}. These bounds are optimal up to the constant in the exponent when c satisfies \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}$\sum_{i=1}^n c_i^2=O(n)$\end{document}; the latter can be viewed as a “finite variance” condition for the child sequence. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Explicit formulas for a sequence of binary tree polynomials

The W-ary tree corresponding to a word W=w0w1⋯wk is the complete planar tree whose nodes at level i have out-degree wi. Investigating some combinatorial aspects of the Shuffle conjecture (Haglund et al. (2005) [1]), Hicks recently discovered (Hicks (2012) [3]) a remarkable family of univariate polynomials associated to W-ary trees. She proved that a single functional equation satisfied by her binary tree polynomials reduces all the compositional cases of the Haglund–Morse–Zabrocki conjectures (Haglund et al. (2011) [2]) to the partition case. This development would entail a substantial advance on the decade-old Shuffle conjecture. We derive here explicit formulas for the Hicks polynomials in the binary tree case, prove her conjecture in this special case and derive some further properties of these binary polynomials that are also conjectured in the thesis work of Hicks.

Sign-balance identities of Adin–Roichman type on 321-avoiding alternating permutations

Adin and Roichman proved a set of refined sign-balance identities on 321-avoiding permutations respecting the last descent of the permutations, which we call the identities of Adin–Roichman type. In this work, we construct a new involution on plane trees that proves refined sign-balance properties on 321-avoiding alternating permutations respecting the first and last entries of the permutations respectively and obtain two sets of identities of Adin–Roichman type.

Divided differences of multivariate implicit functions

Under general conditions, the equation $g(x^1, ..., x^q, y) = 0$ implicitly defines $y$ locally as a function of $x^1, ..., x^q$. In this article, we express divided differences of $y$ in terms of divided differences of $g$, generalizing a recent formula for the case where $y$ is univariate. The formula involves a sum over a combinatorial structure whose elements can be viewed either as polygonal partitions or as plane trees. Through this connection we prove as a corollary a formula for derivatives of $y$ in terms of derivatives of $g$.

The height of watermelons with wall

We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalizes a famous result of de Bruijn et al (1972 Graph Theory and Computing (New York: Academic) pp 15–22) on the average height of planted plane trees, and results by Fulmek (2007 Electron. J. Combin. 14 R64) and Katori et al (2008 J. Stat. Phys. 131 1067–83) on the expected value and higher moments, respectively, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information, we prove a reciprocity relation satisfied by the derivatives of one of Jacobi’s theta functions, which generalizes the well-known reciprocity law for Jacobi’s theta functions.

The expression of the cerato-platanin gene is related to hyphal growth and chlamydospores formation in Ceratocystis platani

Cerato-platanin (CP) is a protein produced by Ceratocystis platani, the causal agent of canker stain disease of plane trees. CP is the first member of the 'cerato-platanin family', and its role as a pathogen-associated molecular pattern (PAMP), inducing defence responses both in host and nonhost plants, is established. However, the primary role of CP and its homologues in the fungal life remains unknown. In the present work, we investigated the regulation of the cp gene during the in vitro growth of C. platani in different conditions and under the effect of potential stress factors. Fungal growth and conidiogenesis were also analysed. Results showed that cp is a single-copy gene whose expression level is strictly associated with hyphal growth and with chlamydospores formation. The analysis of a 1368 bp 5'-flanking region revealed putative motifs that could be involved in the regulation of gene expression in response to stress and developmental cues. Taking into account the localization of CP in the fungal cell wall and the recently published 3D structure of the protein, our results support a role for CP in growth and developmental processes of C. platani.

Neoteny of the European plane tree

The neoteny of 64-day-old European plane tree autovegetative progeny is described, starting from the fact that sexual maturity in the juvenile stage of tree development is a rare and significant process by which the period of synthesis of new lower taxonomic units is shortened. The observed formation of European plane tree reproductive organs has been classified as induced neoteny. Compared to the inflorescences of mature European plane trees, the number of precocious inflorescences was lower, flower structure was simpler, and the diameters of inflorescences were the same. As neoteny is a genetic and physiological marker for the tree genotype and the species, and as it is classified as an example of parallelinherited variability explained by changed interactions of trophic, hormonal and genetic systems, this analysis of neoteny can be a contribution to its further scientific study and its practical utilization at a larger scale.

Indicating atmospheric sulfur by means of S-isotope in leaves of the plane, osmanthus and camphor trees

Foliar δ 34S values of three soil-growing plant species ( Platanus Orientalis L., Osmanthus fragrans L. and Cinnamomum camphora) have been analyzed to indicate atmospheric sulfur. The foliar δ 34S values of the three plant species averaged −3.11 ± 1.94‰, similar to those of both soil sulfur (−3.73 ± 1.04‰) and rainwater sulfate (−3.07 ± 2.74‰). This may indicate that little isotopic fractionation had taken place in the process of sulfur uptake by root or leaves. The δ 34S values changed little in the transition from mature leaves to old/senescing leaves for both the plane tree and the osmanthus tree, suggestive of little isotope effect during sulfur redistribution in plant tissues. Significantly linear correlation between δ 34S values of leaves and rainwater sulfate for the plane and osmanthus trees allowed the tracing of temporal variations of atmospheric sulfur by means of foliar sulfur isotope, while foliage δ 34S values of the camphor is not an effective indicator of atmospheric sulfur.

Random trees with superexponential branching weights

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors wn associated with the vertices of the tree and depending only on their individual degrees n. We focus on the case when wn grows faster than exponentially with n. In this case, the measures on trees of finite size N converge weakly as N tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form wn = ((n − 1)!)α with α > 0, we obtain more refined results about the approach to the infinite volume limit.