Enumeration of General t-ary Trees and Universal Types

We consider t-ary trees characterized by their numbers of nodes and their total path length. When t=2 these are called binary trees, and in such trees a parent node may have up to t child nodes. We give asymptotic expansions for the total number of trees with nodes and path length p, when n and p are large. We consider several different ranges of n and p. For n→∞ and p=O(n^{3/2}) we recover the Airy distribution for the path length in trees with many nodes, and also obtain higher order asymptotic results. For p→∞ and an appropriate range of n we obtain a limiting Gaussian distribution for the number of nodes in trees with large path lengths. The mean and variance are expressed in terms of the maximal root of the Airy function. Singular perturbation methods, such as asymptotic matching and WKB type expansions, are used throughout, and they are combined with more standard methods of analytic combinatorics, such as generating functions, singularity analysis, saddle point method, etc. The results are applicable to problems in information theory, that involve data compression schemes which parse long sequence into shorter phrases. Numerical studies show the accuracy of the various asymptotic approximations. Key Words: Trees; Universal Types; Asymptotics; Path Length; Singular Perturbation

Spatial patterns and processes in a regenerating mangrove forest

The global effort to rehabilitate and restore destroyed mangrove forests is unable to keep up with the high mangrove deforestation rates which exceed the average pace of global deforestation by three to five times. Our knowledge of the underlying processes of mangrove forest regeneration is too limited in order to find suitable techniques for the restoration of degraded mangrove areas. The general objective of my dissertation was to improve mangrove restoration by understanding regeneration processes and local plant-plant interaction in a regenerating Avicennia germinans forest. The study was conducted in a high-shore mangrove forest area on the Ajuruteua peninsula, State of Para, Northern Brazil. The dwarf forest consisting of shrub-like trees is recovering from a stand-replacing event caused by a road construction in 1974 which interrupted the tidal inundation of the study area. Consequently, infrequent inundation and high porewater salinity limit tree growth and canopy closure. All trees and seedlings were stem-mapped in six 20 m x 20 m plots which were located along a tree density gradient. Moreover, height, crown extent, basal stem diameter of trees were measured. The area of herbaceous ground vegetation and wood debris were mapped as well. The mapped spatial distribution of trees, seedlings and covariates was studied using point pattern analysis and point process models, such as Gibbs and Thomas point process, in order to infer underlying ecological processes, such as seed dispersal, seedling establishment, tree recruitment and tree interaction. In the first study (chapter 2), I analyzed the influence of abiotic and biotic factors on the seedling establishment and tree recruitment of A. germinans during the recolonization of severely degraded mangrove sites using point process modeling. Most seedlings established adjacent to adult trees especially under their crown cover. Moreover, seedling density was higher within patches of the herbaceous salt-marsh plants Blutaparon portulacoides and Sesuvium portulacastrum than in uncovered areas. The higher density of recruited A. germinans trees in herb patches indicated that ground vegetation did not negatively influence tree development of A. germinans. In addition, tree recruitment occurred in clusters. Coarse wood debris had no apparent effect on either life stage. These results confirm that salt-marsh vegetation acts as the starting point for mangrove recolonization and indicate that the positive interaction among trees accelerates forest regeneration. In the second study (chapter 3), I analyzed how intraspecific interaction among A. germinans trees determines their growth and size under harsh environmental conditions. Interaction among a higher number of neighboring trees was positively related to the development of a focal tree. However, tree height, internode length and basal stem diameter were only positively associated in low-density forest stands (1.2 trees m-2) and not in forest stands of higher tree density (2.7 trees m-2). These results indicated a shift from facilitation, i.e. a positive effect of tree interaction, towards a balance between facilitation and competition. In the third study (chapter 4), I used point process modeling and the individual-based model mesoFON to disentangle the impact of regeneration and interaction processes on the spatial distribution of seedlings and trees. In this infrequently inundated area, propagules of A. germinans are only dispersed at a maximum distance of 3 m from their parent tree. Furthermore, there is no evidence that the following seedling establishment is influenced by trees. I was able to differentiate positive and negative tree interactions simulated by the mangrove model mesoFON regardless of dispersal processes based on static tree size information using the mark-correlation function. The results of this dissertation suggest that mangrove forest regeneration in degraded areas is a result of facilitative and not competitive interactions among mangrove trees, seedling and herbaceous vegetation. This has important implications for the restoration of degraded mangrove forest. Degraded mangrove areas are usually restored by planting a high number of evenly spaced seedlings. However, high costs constrain this approach to small areas. Assisting natural regeneration could be a less costly alternative. Herbaceous vegetation plays a crucial role in forest recolonization by entrapping propagules and possibly ameliorating harsh environmental conditions. So far only competition among mangrove trees has been considered during restoration. However, facilitative tree interactions could be utilized by planting seedling clusters in order to assist natural regeneration instead of planting seedlings evenly-spaced over large areas. This dissertation also showed that point pattern analysis and point process modeling can enable forest ecologists to describe the spatial distribution of trees as well as to infer underlying ecological processes

Circumscription and phylogeny of the Laurales

The order Laurales comprises a few indisputed core constituents, namely Gomortegaceae, Hernandiaceae, Lauraceae, and Monimiaceae sensu lato, and an equal number of families that have recently been included in, or excluded from, the order, namely Amborellaceae, Calycanthaceae, Chloranthaceae, Idiospermaceae, and Trimeniaceae. In addition, the circumscription of the second largest family in the order, the Monimiaceae, has been problematic. I conducted two analyses, one on 82 rbcL sequences representing all putative Laurales and major lineages of basal angiosperms to clarify the composition of the order and to determine the relationships of the controversal families, and the other on a concatenated matrix of sequences from 28 taxa and six plastid genome regions (rbcL, rpl16, trnT-trnL, trnL-trnF, atpB-rbcL, and psbA-trnH) that together yielded 898 parsimony-informative characters. Fifteen morphological characters that play a key role in the evolution and classification of Laurales were analyzed on the most parsimonious molecular trees as well as being included directly in the analysis in a total evidence approach. The resulting trees strongly support the monophyly of the core Laurales (as listed above) plus Calycanthaceae and Idiospermaceae. Trimeniaceae form a clade with Illiciaceae, Schisandraceae, and Austrobaileyaceae, whereas Amborellaceae and Chloranthaceae represent isolated clades that cannot be placed securely based on rbcL alone. Within Laurales, the deepest split is between Calycanthaceae (including Idiospermaceae) and the remaining six families, which in turn form two clades, the Siparunaceae (Atherospermataceae-Gomortegaceae) and the Hernandiaceae (Monimiaceae s.str. [sensu stricto]-Lauraceae). Monimiaceae clearly are polyphyletic as long as they include Atherospermataceae and Siparunaceae. Several morphological character state changes are congruent with the molecular tree: (1) Calycanthaceae have disulculate tectate-columellate pollen, while their sister clade has inaperturate thin-exined pollen, with the exception of Atherospermataceae, which have columellate but meridionosulcate or disulcate pollen. (2) Calycanthaceae have two ventral ovules while their sister clade has solitary ovules. Within this sister clade, the Hernandiaceae (Lauraceae-Monimiaceae) have apical ovules, while the Siparunaceae (Atherospermataceae-Gomortegaceae) are inferred to ancestrally have basal ovules, a condition lost in Gomortega, the only lauralean genus with a syncarpous ovary. (3) Calycanthaceae lack floral nectaries (except for isolated nectarogeneous fields on the inner tepals), while their sister clade ancestrally has paired nectar glands on the filaments. Filament glands were independently lost in higher Monimiaceae and in Siparunaceae concomitant with pollinator changes away from nectar-foraging flies and bees to non-nectar feeding beetles and gall midges. (4) Disporangiate stamens with anthers dehiscing by two apically hinged valves are ancestral in Siparunaceae-(Atherospermataceae- Gomortegaceae) and evolved independently within Hernandiaceae and Lauraceae. Depending on the correct placement of Calycanthaceae-like fossil flowers, tetrasporangiate anthers with valvate dehiscence (with the valves laterally hinged) may be ancestral in Laurales and lost in modern Calycanthaceae and Monimiaceae

ANALYSIS OF THE GENETIC DIVERSITY OF "LOVRAN MARRON" (Castanea sativa Mill.) USING MICROSATELLITE MARKERS

Maruni (maroni) su sorte europskog pitomog kestena (Castanea sativa Mill.) dobivene selekcijom, koje se od davnina uzgajaju radi proizvodnje krupnih i kvalitetnih plodova. Maruni su u Hrvatskoj sađeni na privatnim posjedima istočnih padina Učke, u okolici Lovrana i poznati su pod nazivom "lovranski marun". Do sada nije bilo znanstvenih istraživanja lovranskog maruna te nije poznato s kojim su biljnim materijalom nasadi podignuti, odnosno koliko je različitih genotipova zastupljeno. Ta saznanja ključna su za sve daljnje korake koje treba poduzeti kako bi se očuvali postojeći genetski izvori. Cilj ovoga istraživanja bila je analiza genetske raznolikosti stabala lovranskog maruna u postojećim nasadima, korištenjem mikrosatelitnih biljega. Istraživanje je rađeno na uzorku od 72 stabla, korištenjem 5 mikrosatelitnih biljega. Analiza je pokazala prisutnost 11 multilokusnih genotipova, što govori u prilog raznovrsnosti i bogatstvu svojti pitomog kestena na lovranskom području, koje još uvijek nisu taksonomski određene, a vode se pod kolektivnim nazivom "lovranski marun". Većina uzorkovanih stabala, 58, pripada istom genotipu, što se može tumačiti statičnošću u smislu introdukcije novih svojti na istraživano područje i forsiranjem, tj. ekstenzivnim uzgojem.Marrons are varieties of the European chestnut (Castanea sativa Mill.) obtained through selection, which have been grown since antiquity for the production of large and high quality fruits. In Croatia, marrons were planted on private properties on the eastern slopes of the Učka mountain, in the environs of Lovran, and are hence known as the "Lovran marron". There has been no scientific research of the Lovran marron to date, and it is unknown which plant material was used to raise the plantations, or how many different genotypes are represented. Those insights are crucial for any further steps to be undertaken in order to conserve the existing genetic resources. The aim of this study was to analyze the genetic diversity of the Lovran marron trees in the existing plantations, by using microsatellite markers. The study was conducted on a sample of 72 trees, using 5 microsatellite markers (Table 1). The analysis demonstrated the presence of 11 multilocus genotypes, pointing to the diversity and abundance of sweet chestnut taxa in the Lovran area, which have not yet been taxonomically defined and bear the collective name of the "Lovran marron". The majority of analyzed trees, specifically 58 individuals, had a uniform genetic structure and areassigned to the MG01 cultivar, which is therefore the most represented cultivar in the researched area, i.e. the one most often grown. However, not all trees are uniform, which is proven by the fact that the remaining 14 analyzed trees belong to 10 different gene pools. Of the 14 trees, 2 had not been grafted, but are found in the plantations together with the grafted marrons and are genetically specific as is to be expected. The remaining 12 grafted trees belong to 9 gene pools. Out of those, 5 trees share common alleles on all loci and are assigned the MG02 cultivar, whereas 7 trees were genetically unique and classified into 7 different cultivars (Tables 2, 3, 4 and Figure 1). Consequently, with regard to the "Lovran marron" operational taxonomic unit grown in the area of the Municipality of Lovran, although it is not taxonomically specified, on the basis of the genetic diversity analysis conducted using 5 microsatellite markers, it can be said to include several different genotypes, or cultivars, one of which (MG01) is present at a much higher frequency than others

Nitrogen forms affect root structure and water uptake in the hybrid poplar

The study analyses the effects of two different forms of nitrogen fertilisation (nitrate and ammonium) on root structure and water uptake of two hybrid poplar (Populus maximowiczii x P. balsamifera) clones in a field experiment. Water uptake was studied using sap flow gauges on individual proximal roots and coarse root structure was examined by excavating 18 whole-root systems. Finer roots were scanned and analyzed for architecture. Nitrogen forms did not affect coarse-root system development, but had a significant effect on fine-root development. Nitrate-treated trees presented higher fine:coarse root ratios and higher specific root lengths than control or ammonium treated trees. These allocation differences affected the water uptake capacity of the plants as reflected by the higher sapflow rate in the nitrate treatment. The diameter of proximal roots at the tree base predicted well the total root biomass and length. The diameter of smaller lateral roots also predicted the lateral root mass, length, surface area and the number of tips. The effect of nitrogen fertilisation on the fine root structure translated into an effect on the functioning of the fine roots forming a link between form (architecture) and function (water uptake)

Molecular phylogenetics of the North American snake tribe Thamnophiini

Advancements in phylogenetic theory and methodology coupled with improvements in computational and sequencing technology facilitate study of the divergence and diversification patterns of life. I apply our current understanding to further explore the relationships and evolution of the North American snake tribe Thamnophiini, as well as to address current topics in phylogenetic and taxonomic methodology. There are two paradigms for the phylogenetic analysis of multi-locus sequence data: one which forces all genes to share the same underlying history, and another that allows genes to follow idiosyncratic patterns of descent from ancestral alleles. The first of these approaches (concatenation) is a simplified model of the actual process of genome evolution while the second (species-tree methods) may be overly complex for histories characterized by long divergence times between cladogenesis. Rather than making an a priori determination concerning which of these phylogenetic models to apply to our data, I seek to provide a framework for choosing between concatenation and species-tree methods that treat genes as independently evolving lineages. In Chapter 2 I demonstrate that parametric bootstrapping can be used to assess the extent to which genealogical incongruence across loci can be attributed to phylogenetic estimation error, and demonstrate the application of our approach using an empirical dataset from 10 species of the Natricine snake sub-family. Since our data exhibit incongruence across loci that is clearly caused by a mixture of coalescent stochasticity and phyogenetic estimation error, we also develop an approach for choosing among species tree estimation methods that take gene trees as input and those that simultaneously estimate gene trees and species trees. Ideally, existing taxonomy would be consistent with phylogenetic estimates derived from rigorously analyzed data using appropriate methods. In Chapter 3 I present a multi-locus molecular analysis of the relationships among nine genera in the North American snake tribe Thamnophiini in order to test the monophyly of the crayfish snakes (genus Regina) and the earth snakes (genus Virginia). Sequence data from seven genes were analyzed to assess relationships among representatives of the nine genera by performing multi-locus phylogeny and species tree estimations, and we performed constraint-based tests of monophyly of classic taxonomic designations on a gene-by-gene basis. Estimates of species trees demonstrate that both genera are paraphyletic, and this inference is supported by a concatenated tree. This finding was supported using gene tree constraint tests and Bayes factors, where we rejected the monophyly of both the crayfish snakes (genus Regina) and the earth snakes (genus Virginia). Progress in our understanding of molecular evolution necessitates a more thorough assessment of the phylogeny of thamnophiine snakes, whose relationships have not been fully resolved, and whose previous phylogenetic estimates are based solely on mitochondrial sequence data. In Chapter 4, I present the most data and taxa robust phylogenetic estimate of Thamnophiini to date, including 50 taxa and sequence data from 8 independently sorting loci. Our findings support the taxonomic recommendations proposed in Chapter 3. Additionally, I estimated the timing of divergence among the three major lineages to have occurred during the Miocene period (~14-11MYA), with higher than expected diversification in the garter snaked during the Pliocene period (~2-6MYA). Finally, we demonstrate that prey choice is labile, and thus an unreliable character for phylogeny reconstruction. Combined, these chapters present a thorough examination of the molecular phylogenetics of thamnophiine snakes. The novel methodological approaches may serve as a guideline for future research. Through estimating a robust phylogeny and suggesting taxonomic changes where appropriate, this work provides a foundation for phylogenetically-based studies of this group

Stochastic Continuous Time Neurite Branching Models with Tree and Segment Dependent Rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p(\gamma, t) for finding a tree \gamma at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number \mu(f(n),t) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p(n,t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.Comment: 41 pages, 2 figures, revised structure and text improvement

Width and mode of the profile for some random trees of logarithmic height

We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quad trees, plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Comparison of Well-Quasi Orders on Trees

Well-quasi orders such as homeomorphic embedding are commonly used to ensure termination of program analysis and program transformation, in particular supercompilation. We compare eight well-quasi orders on how discriminative they are and their computational complexity. The studied well-quasi orders comprise two very simple examples, two examples from literature on supercompilation and four new proposed by the author. We also discuss combining several well-quasi orders to get well-quasi orders of higher discriminative power. This adds 19 more well-quasi orders to the list.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455