We study the local convergence of critical Galton–Watson trees under various conditionings. We give a sufficient condition, which serves to cover all previous known results, for the convergence in distribution of a conditioned Galton–Watson tree to Kesten’s tree. We also propose a new proof to give the limit in distribution of a critical Galton–Watson tree, with finite support, conditioned on having a large width.

Introduction. Leaf area is often related to plant growth, development,physiology and yield. Many non-destructive models have been proposed for leaf areaestimation of several plant genotypes, demonstrating that leaf length, leaf width and leafarea are closely correlated. Thus, the objective of our study was to develop a reliablemodel for leaf area estimation from linear measurements of leaf dimensions for citrusgenotypes. Materials and methods. Leaves of citrus genotypes were harvested,and their dimensions (length, width and area) were measured. Values of leaf area wereregressed against length, width, the square of length, the square of width and the product(length × width). The most accurate equations, either linear or second-order polynomial,were regressed again with a new data set; then the most reliable equation was defined.Results and discussion. The first analysis showed that the variableslength, width and the square of length gave better results in second-order polynomialequations, while the linear equations were more suitable and accurate when the width andthe product (length × width) were used. When these equations were regressed with the newdata set, the coefficient of determination (R2 ) and theagreement index ‘d’ were higher for the one that used the variable product(length × width), while the Mean Absolute Percentage Error was lower.Conclusion. The product of the simple leaf dimensions (length × width) canprovide a reliable and simple non-destructive model for leaf area estimation across citrusgenotypes.

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

Let A be a torsion free abelian group. We say that a group K is a finite essential extension of A if K contains an essential subgroup of finite index which is isomorphic to A. Such K admits a representation as (A ℤ x kx)/ℤky where y = Nx + a for some k x k matrix N over Z and α ∈ A k satisfying certain conditions of relative primeness and ℤk = {(α1,..., αk) : αi, ∈ ℤ}. The concept of absolute width of an f.e.e. K of A is defined and it is shown to be strictly smaller than the rank of A. A kind of basis substitution with respect to Smith diagonal matrices is shown to hold for homogeneous completely decomposable groups. This result together with general properties of our representations are used to provide a self contained proof that acd groups with two critical types are direct sum of groups of rank one and two.