Slicings of parallelogram polyominoes, or how Baxter and Schröder can be reconciled

International audience We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes (called slicings) which grow according to these succession rules. We also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, and a new Schröder subset of Baxter permutations.

Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

Modelling of retinal vasculature based on genetically tuned parametric L-system

Structures of retinal blood vessels are of great importance in diagnosis and treatment of diseases that affect the eyes. Parametric Lindenmayer system (L-system) is one of the powerful rule-based methods that has a great capability for generating tree-like structures using simple rewriting rules. In this study, a novel framework, which can be used to model the retinal vasculature based on available images, has been proposed. This framework presents a solution to special instance of a general open problem, the L-system inverse problem, in which L-system rules should be obtained based on images representing a particular tree-like structure. In this study, genetic algorithm with a novel objective function based on feature matching and an L-system grammar comparison has been used along with nonlinear regression to solve the parametric L-system inverse problem. The resulting L-system growth rules have been employed to predict inaccessible vascular branches. Graphical and quantitative comparison between model results and target structures of different case studies reveals that the proposed framework can be used to generate the structure of retinal blood vessels accurately. Even in the cases lacking sufficient image data, it can provide acceptable predictions.

Management strategies of Eremanthus erythropappus (DC.) MacLeish under different initial spacing

ABSTRACT Eremanthus erythropappus, commonly known as candeia, is an income-generating tree native to Brazil. This is due to the high durability of its wood and its essential oil containing the active component alpha bisabolol. Despite this economic potential, until the early 2000's no studies existed to explore the sustainable management in areas in which the species naturally occurs or for establishing commercial plantations. This study proposes new management strategies based on an individual tree model, and evaluates the growth behavior of candeia trees planted in different spacing. The experiment was installed in March 2002, in Carrancas municipality, Minas Gerais state, Brazil. The experimental area was divided into 4 blocks with 4 different spacings as treatments. The individual model used to propose the best management system uses development of crown area as a function of DBH. The results showed that candeia trees were sensitive to initial spacing variation. With increased initial spacing, candeia trees reached competition later, as demonstrated by crown area development. Thus, candeia trees planted at a wider spacing maintain a desirable growth rate without need for thinning for a longer time, compared to trees planted at narrower spacing. The fitted individual tree model presented in this study showed consistent results and flexibility, providing alternatives for different management strategies. The best growth response was obtained for planting densities greater than 3.75 m² per tree, which corresponds to a spacing of 1.5 x 2.5 m.