219 research outputs found
Width and mode of the profile for some random trees of logarithmic height
We propose a new, direct, correlation-free approach based on central moments
of profiles to the asymptotics of width (size of the most abundant level) in
some random trees of logarithmic height. The approach is simple but gives
precise estimates for expected width, central moments of the width and almost
sure convergence. It is widely applicable to random trees of logarithmic
height, including recursive trees, binary search trees, quad trees,
plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Congruence properties of depths in some random trees
Consider a random recusive tree with n vertices. We show that the number of
vertices with even depth is asymptotically normal as n tends to infinty. The
same is true for the number of vertices of depth divisible by m for m=3, 4 or
5; in all four cases the variance grows linearly. On the other hand, for m at
least 7, the number is not asymptotically normal, and the variance grows faster
than linear in n. The case m=6 is intermediate: the number is asymptotically
normal but the variance is of order n log n.
This is a simple and striking example of a type of phase transition that has
been observed by other authors in several cases. We prove, and perhaps explain,
this non-intuitive behavious using a translation to a generalized Polya urn.
Similar results hold for a random binary search tree; now the number of
vertices of depth divisible by m is asymptotically normal for m at most 8 but
not for m at least 9, and the variance grows linearly in the first case both
faster in the second. (There is no intermediate case.)
In contrast, we show that for conditioned Galton-Watson trees, including
random labelled trees and random binary trees, there is no such phase
transition: the number is asymptotically normal for every m.Comment: 23 page
A functional limit theorem for the profile of -ary trees
In this paper we prove a functional limit theorem for the weighted profile of
a -ary tree. For the proof we use classical martingales connected to
branching Markov processes and a generalized version of the profile-polynomial
martingale. By embedding, choosing weights and a branch factor in a right way,
we finally rediscover the profiles of some well-known discrete time trees.Comment: Published in at http://dx.doi.org/10.1214/09-AAP640 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A simple stochastic model for the evolution of protein lengths
We analyse a simple discrete-time stochastic process for the theoretical
modeling of the evolution of protein lengths. At every step of the process a
new protein is produced as a modification of one of the proteins already
existing and its length is assumed to be random variable which depends only on
the length of the originating protein. Thus a Random Recursive Trees (RRT) is
produced over the natural integers. If (quasi) scale invariance is assumed, the
length distribution in a single history tends to a lognormal form with a
specific signature of the deviations from exact gaussianity. Comparison with
the very large SIMAP protein database shows good agreement.Comment: 12 pages, 4 figure
A functional limit theorem for the profile of search trees
We study the profile of random search trees including binary search
trees and -ary search trees. Our main result is a functional limit theorem
of the normalized profile for in a certain range of . A central feature of the proof is the
use of the contraction method to prove convergence in distribution of certain
random analytic functions in a complex domain. This is based on a general
theorem concerning the contraction method for random variables in an
infinite-dimensional Hilbert space. As part of the proof, we show that the
Zolotarev metric is complete for a Hilbert space.Comment: Published in at http://dx.doi.org/10.1214/07-AAP457 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Shape of Unlabeled Rooted Random Trees
We consider the number of nodes in the levels of unlabelled 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 Galton-Watson
trees and forests to the case of unlabelled rooted trees and show that they
behave in this respect essentially like a conditioned Galton-Watson process.Comment: 34 pages, 1 figur
Fringe trees, Crump-Mode-Jagers branching processes and -ary search trees
This survey studies asymptotics of random fringe trees and extended fringe
trees in random trees that can be constructed as family trees of a
Crump-Mode-Jagers branching process, stopped at a suitable time. This includes
random recursive trees, preferential attachment trees, fragmentation trees,
binary search trees and (more generally) -ary search trees, as well as some
other classes of random trees.
We begin with general results, mainly due to Aldous (1991) and Jagers and
Nerman (1984). The general results are applied to fringe trees and extended
fringe trees for several particular types of random trees, where the theory is
developed in detail. In particular, we consider fringe trees of -ary search
trees in detail; this seems to be new.
Various applications are given, including degree distribution, protected
nodes and maximal clades for various types of random trees. Again, we emphasise
results for -ary search trees, and give for example new results on protected
nodes in -ary search trees.
A separate section surveys results on height, saturation level, typical depth
and total path length, due to Devroye (1986), Biggins (1995, 1997) and others.
This survey contains well-known basic results together with some additional
general results as well as many new examples and applications for various
classes of random trees
Shape Measures of Random Increasing k
International audienceRandom increasing k-trees represent an interesting, useful class of strongly dependent graphs that have been studied widely, including being used recently as models for complex networks. We study in this paper an informative notion called connectivity-profile and derive, by several analytic means, asymptotic estimates for its expected value, together with the limiting distribution in certain cases; some interesting consequences predicting more precisely the shapes of random k-trees are also given. Our methods of proof rely essentially on a bijection between k-trees and ordinary trees, and the resolution of a linear system
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