27 research outputs found
Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees
An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger
Local limits of galton-watson trees conditioned on the number of protected nodes
We consider a marking procedure of the vertices of a tree where each vertex
is marked independently from the others with a probability that depends only on
its out-degree. We prove that a critical Galton-Watson tree conditioned on
having a large number of marked vertices converges in distribution to the
associated size-biased tree. We then apply this result to give the limit in
distribution of a critical Galton-Watson tree conditioned on having a large
number of protected nodes
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