150,864 research outputs found
On weighted depths in random binary search trees
Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg.
Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled
rooted tree is the sum of all labels on the path connecting the node to the
root. We analyze weighted depths of nodes with given labels, the last inserted
node, nodes ordered as visited by the depth first search process, the weighted
path length and the weighted Wiener index in a random binary search tree. We
establish three regimes of nodes depending on whether the second order
behaviour of their weighted depths follows from fluctuations of the keys on the
path, the depth of the nodes, or both. Finally, we investigate a random
distribution function on the unit interval arising as scaling limit for
weighted depths of nodes with at most one child
The shape of random tanglegrams
A tanglegram consists of two binary rooted trees with the same number of
leaves and a perfect matching between the leaves of the trees. We show that the
two halves of a random tanglegram essentially look like two independently
chosen random plane binary trees. This fact is used to derive a number of
results on the shape of random tanglegrams, including theorems on the number of
cherries and generally occurrences of subtrees, the root branches, the number
of automorphisms, and the height. For each of these, we obtain limiting
probabilities or distributions. Finally, we investigate the number of matched
cherries, for which the limiting distribution is identified as well
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
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
Random structures for partially ordered sets
This thesis is presented in two parts. In the first part, we study a family of models
of random partial orders, called classical sequential growth models, introduced by
Rideout and Sorkin as possible models of discrete space-time. We analyse a particular
model, called a random binary growth model, and show that the random partial
order produced by this model almost surely has infinite dimension. We also give
estimates on the size of the largest vertex incomparable to a particular element of
the partial order. We show that there is some positive probability that the random
partial order does not contain a particular subposet. This contrasts with other existing
models of partial orders. We also study "continuum limits" of sequences of
classical sequential growth models. We prove results on the structure of these limits
when they exist, highlighting a deficiency of these models as models of space-time.
In the second part of the thesis, we prove some correlation inequalities for mappings
of rooted trees into complete trees. For T a rooted tree we can define the proportion
of the total number of embeddings of T into a complete binary tree that map the
root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and
Morayne states that, for two binary trees with one a subposet of the other, this
proportion is larger for the larger tree. They conjecture that the same is true for
two arbitrary trees with one a subposet of the other. We disprove this conjecture
by analysing the asymptotics of this proportion for large complete binary trees.
We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a
correlation inequality which enables us to generalise their result in other directions
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