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28
On The Contour Of Random Trees
 SIAM J. Discrete Math
"... 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 preorder traversal of the tree. Using multivariate generating functions and singulari ..."
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Cited by 64 (20 self)
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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 preorder 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.
Basic Analytic Combinatorics of Directed Lattice Paths
 Theoretical Computer Science
, 2001
"... This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then ess ..."
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Cited by 60 (13 self)
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This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.
Search costs in quadtrees and singularity perturbation asymptotics
 Discrete Comput. Geom
, 1994
"... Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a ..."
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Cited by 21 (5 self)
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Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees. The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by intervening bivariate generating functions 1.
Systems Of Functional Equations
 Random Structures & Algorithms
, 1999
"... . The aim of this paper is to discuss the asymptotic properties of the coecients of generating functions which satisfy a system of functional equations. It turns out that under certain general conditions these coecients are related to the distribution of a multivariate random variable that is asy ..."
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Cited by 20 (1 self)
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. The aim of this paper is to discuss the asymptotic properties of the coecients of generating functions which satisfy a system of functional equations. It turns out that under certain general conditions these coecients are related to the distribution of a multivariate random variable that is asymtotically normal. As an application it turns out that the distribution of the terminal symbols in contextfree languages is typically asymptotically normal. 1. Introduction Let Y be a set of combinatorial objects, i.e. every element o 2 Y has a size joj such that the numbers yn = jfo 2 Y : joj = ng are nite for every nonnegative integer n. Especially, if Y has a recursive description then the generating function y(x) = P o2Y x joj = P n0 yn x n satises a functional equation quite frequently. In order to give a rst example and to motivate the topic of this paper let us consider the system of planted plane trees. They can be recursively characterized in the following way: A...
The distribution of nodes of given degree in random trees
 J. Graph Theory
, 1999
"... 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 ..."
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Cited by 15 (6 self)
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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.
Combinatorial Properties of RNA Secondary Structures
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 2001
"... The secondary structure of a RNA molecule is of great importance and possesses inuence, e.g. on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classication of secondary structures by means of their order proved useful with respect to numerous applicati ..."
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Cited by 10 (3 self)
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The secondary structure of a RNA molecule is of great importance and possesses inuence, e.g. on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classication of secondary structures by means of their order proved useful with respect to numerous applications. In 1978 Waterman, who gave the rst precise formal framework for the topic, suggested to determine the number a n;p of secondary structures of size n and given order p. Since then, no satisfactory result has been found. Based on an observation due to Viennot et al. we will derive generating functions for the secondary structures of order p from generating functions for binary tree structures with HortonStrahler number p. These generating functions enable us to compute a precise asymptotic equivalent for a n;p . Furthermore, we will determine the related number of structures when the number of unpaired bases shows up as an additional parameter. Our approach proves to be general enough to compute the average order of a secondary structure together with all the rth moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.
Asymptotic Enumeration via Singularity Analysis
"... Asymptotic formulae for twodimensional arrays (fr,s)r,s≥0 where the associated generating function F (z, w): = � fr,szrw s is meromorphic are provided. Our apr,s≥0 proach is geometrical. To a big extent it generalizes and completes the asymptotic description of the coefficients fr,s along a compac ..."
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Cited by 10 (5 self)
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Asymptotic formulae for twodimensional arrays (fr,s)r,s≥0 where the associated generating function F (z, w): = � fr,szrw s is meromorphic are provided. Our apr,s≥0 proach is geometrical. To a big extent it generalizes and completes the asymptotic description of the coefficients fr,s along a compact set of directions specified by smooth points of the singular variety of the denominator of F (z, w). The scheme we develop can lead to a high level of complexity. However, it provides the leading asymptotic order of fr,s if some unusual and pathological behavior is ruled out. It relies on the asymptotic analysis of a certain type of stationary phase integral of the form � e −s·P (d,θ) A(d, θ)dθ, which describes up to an exponential factor the asymptotic behavior of the coefficients fr,s along the direction d = r s in the (r, s)lattice. The cases of interest are when either the phase term P (d, θ) or the amplitude term A(d, θ) exhibits a change of degree as d approaches a degenerate direction. These are handled by a generalized version of the stationary phase and the coalescing saddle point method which we propose as part of this dissertation. The occurrence of two special functions related to the Airy function is established when two simple saddles of the phase term coalesce. A scheme to study the asymptotic behavior of big powers of generating functions is proposed as an additional application of these generalized methods. ii Dedicated to my mother, father and sister. iii ACKNOWLEDGMENTS I would like to thank to my advisor, Robin Pemantle, for his support and guidance throughout my graduate years at Ohio State. I am deeply indebted for he having supported me as his research assistant for an extended period of time. I also offer my gratitude for his unconditional commitment to connect and keep me in touch with
Images and Preimages in Random Mappings
 SIAM Journal on Discrete Mathematics
, 1996
"... 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 prede ..."
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Cited by 9 (1 self)
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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...
Combinatorics and Asymptotics on Trees
 Cubo Journal
, 2004
"... The purpose of this article is to present explicit and asymptotic methods to count various kinds of trees. In all cases the use of generating functions is essential. Explicit formulae are derived with help of Lagrange's inversion formula. On the other hand singularity analysis of generating ..."
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Cited by 7 (0 self)
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The purpose of this article is to present explicit and asymptotic methods to count various kinds of trees. In all cases the use of generating functions is essential. Explicit formulae are derived with help of Lagrange's inversion formula. On the other hand singularity analysis of generating functions leads to asymptotic formulas.