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13
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 63 (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.
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.
The brownian excursion multidimensional local time density
 Journal of Applied Probability
, 1999
"... Expressions for the multidimensional 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 GaltonWatson trees. ..."
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Cited by 11 (9 self)
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Expressions for the multidimensional 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 GaltonWatson trees.
Stochastic Analysis Of TreeLike Data Structures
 Proc. R. Soc. Lond. A
, 2002
"... The purpose of this article is to present two types of data structures, binary search trees and usual (combinatorial) binary trees. Although they constitute the same set of (rooted) trees they are constructed via completely dierent rules and thus the underlying probabilitic models are dierent, too. ..."
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Cited by 9 (1 self)
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The purpose of this article is to present two types of data structures, binary search trees and usual (combinatorial) binary trees. Although they constitute the same set of (rooted) trees they are constructed via completely dierent rules and thus the underlying probabilitic models are dierent, too. Both kinds of data structures can be analyzed by probabilistic and stochastic tools, binary search trees (more or less) with martingales and binary trees (which can be considered as a special case of GaltonWatson trees) with stochastic processes. It is also an aim of this article to demonstrate the strength of analytic methods in speci c parts of probabilty theory related to combinatorial problems, especially we make use of the concept of generating functions. One reason is that that recursive combinatorial descriptions can be translated to relations for generating functions, and second analytic properties of these generating functions can be used to derive asymptotic (probabilistic) relations. 1.
The Width of GaltonWatson Trees Conditioned by the Size
, 2004
"... It is proved that the moments of the width of GaltonWatson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a ti ..."
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Cited by 7 (1 self)
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It is proved that the moments of the width of GaltonWatson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications. 1.
Strata of random mappings – a combinatorial approach
 Stoch. Proc. Appl
, 1999
"... Abstract. Consider the functional graph of a random mapping from an n–element set into itself. Then the number of nodes in the strata of this graph can be viewed as stochastic process. Using a generating function approach it is shown that a suitable normalization of this process converges weakly to ..."
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Cited by 6 (4 self)
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Abstract. Consider the functional graph of a random mapping from an n–element set into itself. Then the number of nodes in the strata of this graph can be viewed as stochastic process. Using a generating function approach it is shown that a suitable normalization of this process converges weakly to local time of reflecting Brownian bridge. 1.
THE SHAPE OF UNLABELED ROOTED RANDOM TREES
"... Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the h ..."
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Cited by 6 (3 self)
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Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned GaltonWatson trees and forests to the case of unlabeled rooted trees and show that they behave in this respect essentially like a conditioned GaltonWatson process. 1.
On the local time density of the reflecting Brownian bridge
 MR MR1768499 (2001h:60134
, 2000
"... Expressions for the multidimensional 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. ..."
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Cited by 4 (2 self)
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Expressions for the multidimensional 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.
The Width Of GaltonWatson Trees
, 1999
"... . It is proved that the moments of the width of GaltonWatson trees with offspring variance oe are asymptotically given by (oe p n) p mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightne ..."
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Cited by 1 (1 self)
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. It is proved that the moments of the width of GaltonWatson trees with offspring variance oe are asymptotically given by (oe p n) p mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications. 1. Introduction In this paper we are considering rooted trees which are family trees of a GaltonWatson branching process conditioned to have total progeny n. Without loss of generality we may assume that the offspring distribution is given by P f = kg = k ' k '() ; (1) where (' k ; k 0) is a sequence of nonnegative numbers such that '(t) = P k0 ' k t k has a positive or infinite radius of convergence R and is an arbitrary nonnegative number within the circle of convergence of '(t). Due to conditioning it is also no restriction if we confine ourselves to studying only the critical case, i.e., we a...
On the Occupation Time of Brownian Excursion
, 1999
"... Recently, Kalvin M. Jansons derived in an elegant way the Laplace transform of the time spent by an excursion above a given level a>0. This result can also be derived from previous work of the author on the occupation time of the excursion in the interval (a, a + b],bysending b ##. Several alterna ..."
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Cited by 1 (0 self)
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Recently, Kalvin M. Jansons derived in an elegant way the Laplace transform of the time spent by an excursion above a given level a>0. This result can also be derived from previous work of the author on the occupation time of the excursion in the interval (a, a + b],bysending b ##. Several alternative derivations are included. 1 Introduction In [5], the author derives in an elegant way the Laplace transform of the time spent by an excursion above a given level a>0. This result can also be derived from the occupation time of the excursion in the interval (a, a + b], by sending b ##(cf. [2] or [4]). 2 Occupation times Introduce for #, # complex and a # 0, #(#, #, a)= # # cosh(a#)+# sinh(a#) # sinh(a#)+# cosh(a#) # . Denote by W + 0 , Brownian excursion with time parameter t # [0, 1],see[4],I.2foraprecise definition. According to p. 117 and p. 120 of [4], or Theorem 5.1 of [2], the Laplace transform of the occupation time T (a, a + b)= # 1 0 1 (a,a+b] (W + 0 (t)) dt,isg...