90 research outputs found
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
Normal limit laws for vertex degrees in randomly grown hooking networks and bipolar networks
We consider two types of random networks grown in blocks. Hooking networks
are grown from a set of graphs as blocks, each with a labelled vertex called a
hook. At each step in the growth of the network, a vertex called a latch is
chosen from the hooking network and a copy of one of the blocks is attached by
fusing its hook with the latch. Bipolar networks are grown from a set of
directed graphs as blocks, each with a single source and a single sink. At each
step in the growth of the network, an arc is chosen and is replaced with a copy
of one of the blocks. Using P\'olya urns, we prove normal limit laws for the
degree distributions of both networks. We extend previous results by allowing
for more than one block in the growth of the networks and by studying
arbitrarily large degrees.Comment: 28 pages, 6 figure
Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees
We prove limit theorems for sums of functions of subtrees of binary search
trees and random recursive trees. In particular, we give simple new proofs of
the fact that the number of fringe trees of size in the binary search
tree and the random recursive tree (of total size ) asymptotically has a
Poisson distribution if , and that the distribution is
asymptotically normal for . Furthermore, we prove similar
results for the number of subtrees of size with some required property , for example the number of copies of a certain fixed subtree . Using
the Cram\'er-Wold device, we show also that these random numbers for different
fixed subtrees converge jointly to a multivariate normal distribution. As an
application of the general results, we obtain a normal limit law for the number
of -protected nodes in a binary search tree or random recursive tree.
The proofs use a new version of a representation by Devroye, and Stein's
method (for both normal and Poisson approximation) together with certain
couplings
Asymptotic distribution of two-protected nodes in ternary search trees
We study protected nodes in -ary search trees, by putting them in context
of generalised P\'olya urns. We show that the number of two-protected nodes
(the nodes that are neither leaves nor parents of leaves) in a random ternary
search tree is asymptotically normal. The methods apply in principle to -ary search trees with larger as well, although the size of the matrices
used in the calculations grow rapidly with ; we conjecture that the method
yields an asymptotically normal distribution for all .
The one-protected nodes, and their complement, i.e., the leaves, are easier
to analyze. By using a simpler P\'olya urn (that is similar to the one that has
earlier been used to study the total number of nodes in -ary search
trees), we prove normal limit laws for the number of one-protected nodes and
the number of leaves for all
The fluctuations of the giant cluster for percolation on random split trees
A split tree of cardinality is constructed by distributing "balls" in
a subset of vertices of an infinite tree which encompasses many types of random
trees such as -ary search trees, quad trees, median-of- trees,
fringe-balanced trees, digital search trees and random simplex trees. In this
work, we study Bernoulli bond percolation on arbitrary split trees of large but
finite cardinality . We show for appropriate percolation regimes that depend
on the cardinality of the split tree that there exists a unique giant
cluster, the fluctuations of the size of the giant cluster as are described by an infinitely divisible distribution that belongs to
the class of stable Cauchy laws. This work generalizes the results for the
random -ary recursive trees in Berzunza (2015). Our approach is based on a
remarkable decomposition of the size of the giant percolation cluster as a sum
of essentially independent random variables which may be useful for studying
percolation on other trees with logarithmic height; for instance in this work
we study also the case of regular trees.Comment: 43 page
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