Limit laws for functions of fringe trees for binary search trees and random recursive trees

Abstract

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 k = k n in the binary search tree and the random recursive tree (of total size n) asymptotically has a Poisson distribution if k → ∞, and that the distribution is asymptotically normal for k = o( √ n). Furthermore, we prove similar results for the number of subtrees of size k with some required property P , for example 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.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.

Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

“…Furthermore, the results for the m-ary search trees in Section 3.2 (where we consider applications to protected nodes in such trees) are extensions of the results that were proved for the random binary search trees using other methods in [35,19], and extensions of both the results and the methods that were used for m = 2, 3 in [20] and for m = 4, 5, 6 in [18]. The results for the preferential attachment trees in Section 3.3 are extensions of results that previously have been shown for the random recursive trees, see e.g., [19].…”

Cost functionals for large (uniform and simply generated) random trees

“…Thanks to Lemma 2.1, we get that E 1 0 e(s) ds = E[Z 1 ] is finite and that E[Z a−η+1 ] is also finite since a − η + 1 > 1/2 as a > −1/2. Therefore, we deduce from (25) and then (27), (29) and (30) that there exists a finite constant C(a) such that we have for all n ∈ N * : Proof. Let f ∈ B([0, 1]) such that f ≥ 0 and x a f ∞ < +∞ for some a ∈ [0, 1/2).…”

On degenerate sums of m-dependent variables

“…Corollary 1.7 and its reformulation in Remark 1.8 are useful in applications, to show that the asymptotic variance σ 2 > 0. We give two such applications in Section 3, taken from Holmgren and Janson [10] and Janson [15]; these applications were the motivation for the present study. Remark 1.9.…”

“…We sketch two applications of the results above; more details and background are given in Holmgren and Janson [10] and Janson [15]. In both applications we consider a random rooted tree T n with n nodes (with different distributions in the two cases) and let for a fixed rooted tree T , n T (T n ) be the number of nodes v ∈ T n such that the fringe subtree consisting of v and all its descendants is isomorphic to T .…”