Two stochastic processes describing the contour of simply generated random trees are studied: the contour process as defined by Gutjahr and Pflug [9] and the traverse process constructed of the node heights during pre-order traversal of the tree. Using multivariate generating functions and singularity analysis the weak convergence of the contour process to Brownian excursion is shown and a new proof of the analogous result for the traverse process is obtained. 1.

It is shown that all centralized absolute moments EjHn EHn j ( 0) of the height Hn of binary search trees of size n and of the saturation level H 0 n are bounded. The methods used rely on the analysis of a retarded differential equation of the form 0 (u) = 2 (u= ) 2 with > 1. The method can also be extended to prove the same result for the height of m-ary search trees. Finally a conjecture for the distribution of the height is presented.

This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the average-case analysis of algorithms. It develops a general approach to the distributional analysis of parameters of elementary combinatorial structures like strings, trees, graphs, permutations, and so on. The methods are essentially analytic and relie on multivariate generating functions, singularity analysis, and continuity theorems. The limit laws that are derived mostly belong to the Gaussian, Poisson, or geometric type.

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponential-cubic type (e ix 3 ), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when conuences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and ne optimization of random generation algorithms for multiply connected planar graphs.

Abstract. Let Tn denote the set of unrooted unlabeled trees of size n and let k ≥ 1 be given. By assuming that every tree of Tn is equally likely it is shown that the limiting distribution of the number of nodes of degree k is normal with mean value ∼ µkn and variance ∼ σ2 kn with positive constants µk and σk. Besides, the asymptotic behavior of µk and σk for k → ∞ as well as the corresponding multivariate distributions are derived. Furthermore, similar results can be proved for plane trees, for labeled trees, and for forests. 1.

Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for Galton-Watson trees.

Several new asymptotic estimates (with precise error bounds) are derived for Poisson and binomial distributions as the parameters tend to infinity. The analytic methods used are also applicable to other discrete distribution functions.

We present a general theorem which can be used to identify the limiting distribution for a class of combinatorial schemata. For example, many parameters in random mappings can be covered in this way. Especially we can derive the limiting distibution of those points with a given number of total predecessors. 1 Introduction By a random mapping ' 2 Fn F = S n0 Fn we mean an arbitrary mapping ' : f1; : : : ; ng ! f1; : : : ; ng such that every mapping has equal probability n n . The main purpose of this paper is to obtain limit theorems, when n tends to innity, for special parameters in random mappings, e.g. for the number of image points. Since every random mapping ' 2 Fn has equal probability it suces to count the number of radom mappings ' 2 Fn satisfying a special property, e.g. that the number of image points equals k. By dividing this number by n n we get the probability of interest. In order to get the limit distribution for n !1 it is not necessary to know the exact v...

Abstract. An approach via generating functions is used to derive multivariate asymptotic distributions for the number of nodes in strata of random forests. For a certain range for the strata numbers we obtain a weak limit theorem to Brownian motion as well. Moreover, a moment convergence theorem for the width of random forests is derived. 1.

Expressions for the multi-dimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings.