Limit laws for functions of fringe trees for binary search trees and random recursive trees
2015 (English)In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, 4Article in journal (Refereed) Published
We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. 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. As a consequence, we give simple new proofs of the fact that the number of fringe trees of size k = k(n) in the binary search tree or in the random recursive tree (of total size n) has an asymptotical Poisson distribution if k -> infinity, and that the distribution is asymptotically normal for k = o(root n). Furthermore, we prove similar results for the number of subtrees of size k with some required property P, e.g., the number of copies of a certain fixed subtree T. Using the Cramer-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of l-protected nodes in a binary search tree or in a random recursive tree.
Place, publisher, year, edition, pages
2015. Vol. 20, 4
Fringe trees, Stein's method, Couplings, Limit laws, Binary search trees, Random recursive trees
Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:uu:diva-248646DOI: 10.1214/EJP.v20-3627ISI: 000350286000001OAI: oai:DiVA.org:uu-248646DiVA: diva2:802290