Abstract
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\rightarrow\infty $, and that the distribution is asymptotically normal for $ k=o(\sqrt{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 Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. <br /><br />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 $ \ell $-protected nodes in a binary search tree or in a random recursive tree.
Highlights
In this paper we consider fringe trees of the random binary search tree as well as of the random recursive tree; recall that a fringe tree is a subtree consisting of some node and all its descendants, see Aldous [1] for a general theory, and note that fringe trees typically are "small" compared to the whole tree. (All subtrees considered in the present paper are of this type, and we will use ’subtree’ and ’fringe tree’ as synonyms.) We will use a representation of Devroye [10, 11] for the random binary search tree, and a well-known bijection between binary trees and recursive trees, together with different
Limit laws for functions of fringe trees applications of Stein’s method for both normal and Poisson approximation to give both new general results on the asymptotic distributions for random variables depending on fringe trees, and more direct proofs of several earlier results in the field
We state the main results : More precisely, we identify two main results that we present in Section 1.3, one on Poisson approximation and one on normal approximation, for general functions of fringe trees for the random binary search tree, as well as for the random recursive tree
Summary
In this paper we consider fringe trees of the random binary search tree as well as of the random recursive tree; recall that a fringe tree is a subtree consisting of some node and all its descendants, see Aldous [1] for a general theory, and note that fringe trees typically are "small" compared to the whole tree. (All subtrees considered in the present paper are of this type, and we will use ’subtree’ and ’fringe tree’ as synonyms.) We will use a representation of Devroye [10, 11] for the random binary search tree, and a well-known bijection between binary trees and recursive trees, together with different. (All subtrees considered in the present paper are of this type, and we will use ’subtree’ and ’fringe tree’ as synonyms.) We will use a representation of Devroye [10, 11] for the random binary search tree, and a well-known bijection between binary trees and recursive trees, together with different. Limit laws for functions of fringe trees applications of Stein’s method for both normal and Poisson approximation to give both new general results on the asymptotic distributions for random variables depending on fringe trees, and more direct proofs of several earlier results in the field. We state the main results : More precisely, we identify two main results that we present, one on Poisson approximation and one on normal approximation, for general functions of fringe trees for the random binary search tree, as well as for the random recursive tree. We give some examples of applications of the general results on functions of fringe trees in Section 8, for example to the number of protected nodes in the random binary search tree or in the random recursive tree studied by Mahmoud and Ward [35, 36] and to the shape functionals for the random binary search tree or the random recursive tree
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