16,320 research outputs found
The height of random -trees and related branching processes
We consider the height of random k-trees and k-Apollonian networks. These
random graphs are not really trees, but instead have a tree-like structure. The
height will be the maximum distance of a vertex from the root. We show that
w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to
clog t, where t is the number of vertices, and c=c(k) is given as the solution
to a transcendental equation. The equations are slightly different for the two
types of process. In the limit as k-->oo the height of both processes is
asymptotic to log t/(k log 2)
Random real trees
We survey recent developments about random real trees, whose prototype is the
Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain
the formalism of real trees, which yields a neat presentation of the theory and
in particular of the relations between discrete Galton-Watson trees and
continuous random trees. We then discuss the particular class of self-similar
random real trees called stable trees, which generalize the CRT. We review
several important results concerning stable trees, including their branching
property, which is analogous to the well-known property of Galton-Watson trees,
and the calculation of their fractal dimension. We then consider spatial trees,
which combine the genealogical structure of a real tree with spatial
displacements, and we explain their connections with superprocesses. In the
last section, we deal with a particular conditioning problem for spatial trees,
which is closely related to asymptotics for random planar quadrangulations.Comment: 25 page
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
The coalescent point process of branching trees
We define a doubly infinite, monotone labeling of Bienayme-Galton-Watson
(BGW) genealogies. The genealogy of the current generation backwards in time is
uniquely determined by the coalescent point process , where
is the coalescence time between individuals i and i+1. There is a Markov
process of point measures keeping track of more ancestral
relationships, such that is also the first point mass of . This
process of point measures is also closely related to an inhomogeneous spine
decomposition of the lineage of the first surviving particle in generation h in
a planar BGW tree conditioned to survive h generations. The decomposition
involves a point measure storing the number of subtrees on the
right-hand side of the spine. Under appropriate conditions, we prove
convergence of this point measure to a point measure on
associated with the limiting continuous-state branching (CSB) process. We prove
the associated invariance principle for the coalescent point process, after we
discretize the limiting CSB population by considering only points with
coalescence times greater than .Comment: Published in at http://dx.doi.org/10.1214/11-AAP820 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations
We consider a (sub) critical Galton-Watson process with neutral mutations
(infinite alleles model), and decompose the entire population into clusters of
individuals carrying the same allele. We specify the law of this allelic
partition in terms of the distribution of the number of clone-children and the
number of mutant-children of a typical individual. The approach combines an
extension of Harris representation of Galton-Watson processes and a version of
the ballot theorem. Some limit theorems related to the distribution of the
allelic partition are also given.Comment: This version corrects a significant mistake in the first on
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