- Open Access
RETRACTED ARTICLE: Strong limiting behavior in binary search trees
© Chen and Peng; licensee Springer 2013
- Received: 14 June 2012
- Accepted: 6 November 2012
- Published: 20 February 2013
The Retraction Note to this article has been published in Journal of Inequalities and Applications 2013 2013:416
In a binary search tree of size n, each node has no more than two children, we denote the number of the node with k children by . In this paper, we study the strong limit behavior of the random variables and , where represents the number of subtrees of size m. The results can imply some known results.
MSC: 60F05, 05C80.
- binary search tree
- strong limit properties
A binary tree is either empty or composed of a root node together with left and right subtrees which are themselves binary trees. A binary search tree T for a set of keys from a total order, say , is a binary tree in which each node has a key value and all the keys of the left subtree are less than the key at the root, and all the keys of the right subtree are greater than the key at the root, i.e., the first key is associated with the root, the next key is placed in the left child of the root if it is smaller than the key of the root and it is sent to the right child of the root if it is larger than the key of the root. In this way, we proceed further by inserting key by key. This property holds recursively for the left and right subtrees of the tree T.
Usually, it is assumed that every permutation of is equally likely and has the same probability . Hence, any parameter of the binary search trees may be considered as a random variable.
The binary tree model turns out to be appropriate in formal language theory, computer algebra, etc., whereas the binary search tree model is of importance in sorting and searching algorithms and a lot of combinatorial algorithms. See, e.g., [1–4] for a detailed description. There are several papers devoted to the study of properties of the parameters. Kirschenhofer (see ) considered the height of leaves; Panholzer and Prodinger (see ) studied the number of ascendants and the number of descendants of any fixed node. Devroye and Neininger (see ) obtained the tail bounds and the order of higher moments for the path distance between any couples of nodes. Mahmoud and Neininger (see ) arrived at a Gaussian limit law for the distance between randomly selected pairs of nodes in random binary search trees and identified the rate of convergence. Svante (see ) got the exact and asymptotic formulas for moments and an asymptotic distribution for the difference between the left and right total pathlenghts. Devroye (see ) analyzed the properties of some parameter in binary search trees by applying the Stein’s method. There were also many authors, who were interested in the height of binary search trees and drew a variety of properties such as the asymptotic expected value, the variance and the limiting distribution of the height (see [11–20]). Prodinger (see ) computed the probability that a random binary tree with n nodes had i nodes with 2 children. Rote (see ) gave three combinatorial proofs for the number of the binary trees having a given number of nodes with 0, 1, and 2 children. Liu et al. (see ) have studied the limiting theorems for the nodes in binary search trees. Su et al. (see ) have studied some limit properties on the subtrees of random binary search trees.
Let denote a random binary search tree of size n. In the binary search tree, every node has two children at most, we denote the number of the node with k () children by . Let be the number of subtrees of size m in . In , Liu et al. have studied the limit properties of and in the sense of probably. In this paper, by computing the exact expression of the fourth moment of and , we obtain the strong limit properties (in the sense of a.e.) of them by the Chebyshev inequality and the Borel-Cantelli lemma. Obviously, the results can imply the case in .
Lemma 1 (see )
where , and the two random variables and are the independent conditioning on .
Lemma 2 (see )
where EX and VarX denote the expectation and variance of X, respectively.
In the following, by computing the fourth moment of and , we obtain the strong limit properties (in the sense of a.e.) of them by the Chebyshev inequality and the Borel-Cantelli lemma.
By (15), we have (4) holds in the case ; by (13), (5), and (6), we have (4) holds in the case and . This completes the proof. □
In the following, we will show the strong limit property of in a random binary search tree of size n.
By (24), (16) holds. This is the end of the proof. □
This work is supported by Anhui Scientific Research Foundation (KJ2012A205, KJ2012B117, KJ2013B162) and Chaohu University Scientific Research Foundation.
- Kemp R: Fundamentals of the Average Case Analysis of Particular Algorithms. Wiley-Teubner, Stuttgart; 1984.View ArticleMATHGoogle Scholar
- Knuth D 3. In The Art of Computer Programming: Sorting and Searching. Addison-Wesley, Reading; 1973.Google Scholar
- Mahmoud H: Evolution of Random Search Trees. Wiley, New York; 1992.MATHGoogle Scholar
- Sedgewick R, Flajolet P: An Introduction to the Analysis of Algorithms. Addison-Wesley, Reading; 1996.MATHGoogle Scholar
- Kirschenhofer P: On the height of leaves in binary trees. J. Comb. Inf. Syst. Sci. 1983, 8: 44–60.MATHGoogle Scholar
- Panholzer A, Prodinger H: Descendants and ascendants in binary trees. Discrete Math. Theor. Comput. Sci. 1997, 1: 247–266.MATHGoogle Scholar
- Devroye L, Neininger R: Distances and finger search in random binary search trees. SIAM J. Comput. 2004, 33: 647–658. 10.1137/S0097539703424521View ArticleMATHGoogle Scholar
- Mahmoud H, Neininger R: Distribution of distances in random binary search trees. Ann. Appl. Probab. 2003, 13(1):253–276.View ArticleMATHGoogle Scholar
- Svante J: Left and right pathlenghts in random binary trees. Algorithmica 2006, 46: 419–429. 10.1007/s00453-006-0099-3View ArticleMATHGoogle Scholar
- Devroye L: Applications of Stein’s method in the analysis of random binary search trees. Institute for Mathematical Sciences Lecture Notes Series 5. In Stein’s Method and Applications. Edited by: Chen L, Barbour A. World Scientific, Singapore; 2005:247–297.View ArticleGoogle Scholar
- Drmota M: An analytic approach to the height of binary search trees. J. Assoc. Comput. Mach. 2001, 29: 89–119.MATHGoogle Scholar
- Drmota M: An analytic approach to the height of binary search trees II. J. Assoc. Comput. Mach. 2003, 50: 333–374. 10.1145/765568.765572View ArticleMATHGoogle Scholar
- Devroye L: A note on the height of binary search trees. J. Assoc. Comput. Mach. 1986, 33: 489–498. 10.1145/5925.5930View ArticleMATHGoogle Scholar
- Devroye L: Branching processes in the analysis of the height of trees. Acta Inform. 1987, 24: 277–298.MATHGoogle Scholar
- Drmota M: The variance of the height of binary search trees. Theor. Comput. Sci. 2002, 270: 913–919. 10.1016/S0304-3975(01)00006-8View ArticleMATHGoogle Scholar
- Flajolet P, Martinez C, Gourdon X: Patterns in random binary search trees. Random Struct. Algorithms 1997, 11: 223–244. 10.1002/(SICI)1098-2418(199710)11:3<223::AID-RSA2>3.0.CO;2-2View ArticleMATHGoogle Scholar
- Robson JM: On the concentration of the height of binary search trees. Lecture Notes in Computer Science 1256. ICALP 97 Proceedings 1997, 441–448.Google Scholar
- Robson JM: Constant bounds on the moments of the height of binary search trees. Theor. Comput. Sci. 2002, 276: 435–444. 10.1016/S0304-3975(01)00306-1View ArticleMATHGoogle Scholar
- Su C, Liu J, Feng QQ: A note on the distance in random recursive trees. Stat. Probab. Lett. 2006, 76: 1748–1755. 10.1016/j.spl.2006.04.020View ArticleMATHGoogle Scholar
- Su C, Liu J, Hu ZS: The law of large numbers of the size of complete interval trees. Adv. Math. China 2007, 36(2):181–188. (in Chinese)Google Scholar
- Prodinger H: A note on the distribution of the three types of nodes in uniform binary trees. Sémin. Lothar. Comb. 1996., 38: Article ID 38b (electronic)Google Scholar
- Rote G: Binary trees having a given number of nodes with 0, 1, and 2 children. Sémin. Lothar. Comb. 1997., 38: Article ID 38b (electronic)Google Scholar
- Liu J, Su C, Chen Y: Limiting barticle for the nodes in binary search trees. Sci. China Ser. A, Math. 2008, 51(1):101–114. 10.1007/s11425-007-0129-xView ArticleMATHGoogle Scholar
- Su C, Miao BQ, Feng QQ: On the subtrees of random binary search trees. Chin. J. Appl. Probab. Stat. 2006, 22(3):304–310. (in Chinese)MATHGoogle Scholar
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