RETRACTED ARTICLE: Strong limiting behavior in binary search trees

Peishu Chen; Weicai Peng

doi:10.1186/1029-242X-2013-60

Research
Open Access

RETRACTED ARTICLE: Strong limiting behavior in binary search trees

Peishu Chen¹Email author and
Weicai Peng¹

Journal of Inequalities and Applications20132013:60

https://doi.org/10.1186/1029-242X-2013-60

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

Abstract

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 $ξ_{n, k}$ . In this paper, we study the strong limit behavior of the random variables $ξ_{n, k}$ and $σ_{n, m}$ , where $σ_{n, m}$ represents the number of subtrees of size m. The results can imply some known results.

MSC: 60F05, 05C80.

Keywords

binary search tree
strong limit properties
nodes
subtrees

1 Introduction

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 ${1, 2, \dots, n}$ , 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 ${1, 2, \dots, n}$ is equally likely and has the same probability $1 / n!$ . 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 [5]) considered the height of leaves; Panholzer and Prodinger (see [6]) studied the number of ascendants and the number of descendants of any fixed node. Devroye and Neininger (see [7]) obtained the tail bounds and the order of higher moments for the path distance between any couples of nodes. Mahmoud and Neininger (see [8]) 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 [9]) got the exact and asymptotic formulas for moments and an asymptotic distribution for the difference between the left and right total pathlenghts. Devroye (see [10]) 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 [21]) computed the probability that a random binary tree with n nodes had i nodes with 2 children. Rote (see [22]) 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 [23]) have studied the limiting theorems for the nodes in binary search trees. Su et al. (see [24]) have studied some limit properties on the subtrees of random binary search trees.

Let $T_{n}$ 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 ( $= 0, 1, 2$ ) children by $ξ_{n, k}$ . Let $σ_{n, m}$ be the number of subtrees of size m in $T_{n}$ . In [23], Liu et al. have studied the limit properties of $ξ_{n, k}$ and $σ_{n, m}$ in the sense of probably. In this paper, by computing the exact expression of the fourth moment of $ξ_{n, k}$ and $σ_{n, m}$ , 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 [23].

In

T_{n}

, we have

σ_{n, 1} = ξ_{n, 0}

,

σ_{n, k} = 0

as

n < k

, and

σ_{k, k} = 1

. Let

L_{n}

and

R_{n}

be the size of the left and right subtrees of

T_{n}

. It is clear that

L_{n} + R_{n} = n - 1

. From the procession of constructing binary search trees, we can see that

L_{n}

and

R_{n}

are two random variables with uniform distribution on

{0, 1, 2, \dots, n - 1}

. We write

X \overset{D}{=} Y

if X and Y have the same distribution. It is easy to find that

R_{n} = n - 1 - L_{n} \overset{D}{=} L_{n} .

In the following, we will show several properties of $ξ_{n, k}$ , and $σ_{n, m}$ , the proofs can be seen in [23] and [24].

Lemma 1 (see [23])

Let

ξ_{n, 0}

be the number of nodes having no children in a binary search tree of size n. For any positive integer

n \geq 2

, we have

ξ_{n, 0} \overset{D}{=} ξ_{L_{n}, 0} + ξ_{n - 1 - L_{n}, 0}^{*},

(1)

where $ξ_{L_{n}, 0} \overset{D}{=} ξ_{n - 1 - L_{n}, 0}^{*}$ , and the two random variables $ξ_{L_{n}, 0}$ and $ξ_{n - 1 - L_{n}, 0}^{*}$ are the independent conditioning on $L_{n}$ .

Lemma 2 (see [23])

In a binary search tree of size n, when

n \geq 3

, we have

E ξ_{n, 0} = E ξ_{n, 1} = \frac{n + 1}{3}, E ξ_{n, 2} = \frac{n - 2}{3},

and

Var ξ_{n, 0} = Var ξ_{n, 2} = \frac{n + 1}{18}, Var ξ_{n, 1} = \frac{2 (n + 1)}{9} .

When

n > m

,

E σ_{n, m} = \frac{2 (n + 1)}{(m + 1) (m + 2)};

(2)

when

n > 2 m

,

Var σ_{n, m} = \frac{m (5 m - 2) (n + 1)}{(m + 1) {(m + 2)}^{2} (2 m + 1)} : = (n + 1) d_{m}^{2},

(3)

where EX and VarX denote the expectation and variance of X, respectively.

In the following, by computing the fourth moment of $ξ_{n, k}$ and $σ_{n, m}$ , we obtain the strong limit properties (in the sense of a.e.) of them by the Chebyshev inequality and the Borel-Cantelli lemma.

2 Main result

Theorem 1 In a binary search tree of size n, for any integer

k = 0, 1, 2

, we have

lim_{n \to \infty} \frac{ξ_{n, k}}{n} = \frac{1}{3} a.e.

(4)

Proof From the Theorem 5 of [23], we already have

ξ_{n, 0} = ξ_{n, 2} + 1

and

ξ_{n, 0} + ξ_{n, 1} + ξ_{n, 2} = n

. Then we have by Lemma 2

E {[ξ_{n, 1} - \frac{n + 1}{3}]}^{4} = E {[n + 1 - 2 ξ_{n, 0} - \frac{n + 1}{3}]}^{4} = 16 E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4},

(5)

E {[ξ_{n, 2} - \frac{n - 2}{3}]}^{4} = E {[ξ_{n, 0} - 1 - \frac{n - 2}{3}]}^{4} = E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4} .

(6)

In the following, we will show the exact expression of

E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4}

, then we can get (4) by the Borel-Cantelli lemma.

\begin{array}{l} E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4} \\ = E {[ξ_{L_{n}, 0} + ξ_{n - 1 - L_{n}, 0}^{*} - \frac{n + 1}{3}]}^{4} \\ = E [\sum_{j = 0}^{n - 1} {(ξ_{j, 0} - \frac{j + 1}{3} + ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3})}^{4} P (L_{n} = j)] \\ = \frac{1}{n} \sum_{j = 0}^{n - 1} E {[ξ_{j, 0} - \frac{j + 1}{3} + ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3}]}^{4} \\ = \frac{1}{n} \sum_{j = 0}^{n - 1} {E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{4} + 4 E [{(ξ_{j, 0} - \frac{j + 1}{3})}^{3} (ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3})] \\ + 6 E [{(ξ_{j, 0} - \frac{j + 1}{3})}^{2} {(ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3})}^{2}] + 4 E [(ξ_{j, 0} - \frac{j + 1}{3}) {(ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3})}^{3}] \\ + E {[ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3}]}^{4}} . \end{array}

(7)

By the independence of

ξ_{j, 0}

and

ξ_{n - j - 1, 0}^{*}

, and

ξ_{j, 0} \overset{D}{=} ξ_{n - j - 1, 0}^{*}

, we have

E [{(ξ_{j, 0} - \frac{j + 1}{3})}^{3} (ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3})] = E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{3} E [ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3}] = 0,

(8)

E [(ξ_{j, 0} - \frac{j + 1}{3}) {(ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3})}^{3}] = E [ξ_{j, 0} - \frac{j + 1}{3}] E {[ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3}]}^{3} = 0 .

(9)

By

ξ_{j, 0} \overset{D}{=} ξ_{n - j - 1, 0}^{*}

, (7), (8), and (9), we have

\begin{array}{rcl} E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4} & = & \frac{2}{n} \sum_{j = 0}^{n - 1} E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{4} + \frac{6}{n} \sum_{j = 0}^{n - 1} E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{2} E {[ξ_{n - j - 1, 0}^{*} - \frac{n - j}{3}]}^{2} \\ = & \frac{2}{n} \sum_{j = 0}^{n - 1} E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{4} + \frac{6}{n} \sum_{j = 0}^{n - 1} \frac{(j + 1) (n - j)}{18^{2}} . \end{array}

(10)

Using (10) once again, we arrive at

\begin{array}{rcl} E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4} & = & \frac{2}{n} \sum_{j = 0}^{n - 2} E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{4} + \frac{2}{n} E {[ξ_{n - 1, 0} - \frac{n}{3}]}^{4} + \frac{6}{n} \sum_{j = 0}^{n - 1} \frac{(j + 1) (n - j)}{18^{2}} \\ = & \frac{2}{n} \sum_{j = 0}^{n - 2} E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{4} + \frac{6}{n} \sum_{j = 0}^{n - 2} \frac{(j + 1) (n - 1 - j)}{18^{2}} + \frac{2}{n} E {[ξ_{n - 1, 0} - \frac{n}{3}]}^{4} \\ - \frac{6}{n} \sum_{j = 0}^{n - 2} \frac{(j + 1) (n - 1 - j)}{18^{2}} + \frac{6}{n} \sum_{j = 0}^{n - 1} \frac{(j + 1) (n - j)}{18^{2}} \\ = & \frac{n - 1}{n} [\frac{2}{n - 1} \sum_{j = 0}^{n - 2} E {[ξ_{j, 0} - \frac{j + 1}{3}]}^{4} + \frac{6}{n - 1} \sum_{j = 0}^{n - 2} \frac{(j + 1) (n - 1 - j)}{18^{2}}] \\ + \frac{2}{n} E {[ξ_{n - 1, 0} - \frac{n}{3}]}^{4} + \frac{6}{n} \sum_{j = 0}^{n - 2} \frac{j + 1}{18^{2}} + \frac{1}{54} \\ = & \frac{n + 1}{n} E {[ξ_{n - 1, 0} - \frac{n}{3}]}^{4} + \frac{n + 1}{108} . \end{array}

Hence, we have

\frac{E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4}}{n + 1} = \frac{E {[ξ_{n - 1, 0} - \frac{n}{3}]}^{4}}{n} + \frac{1}{108} .

(11)

By (11), we obtain

\frac{E {[ξ_{n, 0} - \frac{n + 1}{3}]}^{4}}{n + 1} = \dots = \frac{E {[ξ_{2, 0} - 1]}^{4}}{3} + \frac{n - 2}{108} = \frac{n - 2}{108} .

(12)

By (12),

E {[\frac{ξ_{n, 0}}{n + 1} - \frac{1}{3}]}^{4} = \frac{n - 2}{108 {(n + 1)}^{3}} \leq \frac{1}{108 {(n + 1)}^{2}} .

(13)

By the Chebyshev inequality and (13), for arbitrary

ε > 0

, we have

P (| \frac{ξ_{n, 0}}{n + 1} - \frac{1}{3} | \geq ε) \leq \frac{E {[\frac{ξ_{n, 0}}{n + 1} - \frac{1}{3}]}^{4}}{ε^{4}} \leq \frac{1}{108 {(n + 1)}^{2} ε^{4}} .

Then

\sum_{n = 1}^{\infty} P (| \frac{ξ_{n, 0}}{n + 1} - \frac{1}{3} | \geq ε) < \infty .

(14)

By the Borel-Cantelli lemma and (14), for arbitrary

ε > 0

, we have

P (⋃_{n \geq k} {| \frac{ξ_{n, 0}}{n + 1} - \frac{1}{3} | \geq ε}) \to 0 (k \to \infty) .

(15)

By (15), we have (4) holds in the case $k = 0$ ; by (13), (5), and (6), we have (4) holds in the case $k = 1$ and $k = 2$ . This completes the proof. □

In the following, we will show the strong limit property of $σ_{n, m}$ in a random binary search tree of size n.

Theorem 2 Let n, m be two positive integers,

σ_{n, m}

be the number of subtrees of size m (<n) in random binary search trees of size n. Then

lim_{n \to \infty} \frac{σ_{n, m}}{n} = \frac{2}{(m + 1) (m + 2)} a.e.

(16)

Proof When

j < m

,

σ_{j, m} \equiv 0

;

σ_{m, m} = 1

; when

n > m

, by (2) and (3), we have

\begin{array}{l} E {[σ_{n, m} - \frac{2 (n + 1)}{(m + 1) (m + 2)}]}^{4} \\ = E {[(σ_{L_{n}, m} - \frac{2 (L_{n} + 1)}{(m + 1) (m + 2)}) + (σ_{n - 1 - L_{n}, m}^{*} - \frac{2 (R_{n} + 1)}{(m + 1) (m + 2)})]}^{4} \\ = \frac{2}{n} \sum_{j = n - m}^{n - 1} E {[(σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}) + (0 - \frac{2 (n - j)}{(m + 1) (m + 2)})]}^{4} \\ + \frac{1}{n} \sum_{j = m}^{n - m - 1} E {[(σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}) + (σ_{n - 1 - j, m}^{*} - \frac{2 (n - j)}{(m + 1) (m + 2)})]}^{4} \\ = \frac{2}{n} \sum_{j = m}^{n - 1} E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{4} \\ + \frac{6}{n} \sum_{j = m}^{n - m - 1} E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{2} E {[σ_{n - 1 - j, m}^{*} - \frac{2 (n - j)}{(m + 1) (m + 2)}]}^{2} \\ - \frac{8}{n} \sum_{j = n - k}^{n - 1} \frac{2 (n - j)}{(k + 1) (k + 2)} E {[σ_{j, k} - \frac{2 (j + 1)}{(k + 1) (k + 2)}]}^{3} \\ + \frac{12}{n} \sum_{j = n - m}^{n - 1} {[\frac{2 (n - j)}{(m + 1) (m + 2)}]}^{2} E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{2} \\ + \frac{2}{n} \sum_{j = n - m}^{n - 1} {[\frac{2 (n - j)}{(m + 1) (m + 2)}]}^{4} \\ = \frac{n + 1}{n} E {[σ_{n - 1, m} - \frac{2 n}{(m + 1) (m + 2)}]}^{4} + \frac{6 d_{m}^{4}}{n} {\sum_{j = m}^{n - 1} (j + 1) (n - j) - \sum_{j = m}^{n - 2} (j + 1) (n - 1 - j)} \\ - \frac{16}{n (m + 1) (m + 2)} {\sum_{j = n - m}^{n - 1} (n - j) E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{3} \\ - \sum_{j = n - 1 - m}^{n - 2} (n - 1 - j) E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{3}} \\ + \frac{12 d_{m}^{2}}{n} {\sum_{j = n - m}^{n - 1} {[\frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{2} (j + 1) - \sum_{j = n - 1 - m}^{n - 2} {[\frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{2} (j + 1)} . \end{array}

(17)

The second term of (17),

\begin{array}{rcl} \frac{6 d_{m}^{4}}{n} {\sum_{j = m}^{n - 1} (j + 1) (n - j) - \sum_{j = m}^{n - 2} (j + 1) (n - 1 - j)} & = & 3 (n - 2 m - 1) d_{m}^{4} + \frac{6 (n - m) (m + 1)}{n} d_{m}^{4} \\ = & O (n + 1), \end{array}

(18)

where

x_{n} = O (n)

means that

x_{n} / n \to constant

as

n \to \infty

. Then by

σ_{n - 1 - m, m} / n \leq 1

, the third term of (17),

\begin{array}{l} \frac{16}{n} {\sum_{j = n - 1 - m}^{n - 2} \frac{n - 1 - j}{(m + 1) (m + 2)} E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{3} \\ - \sum_{j = n - m}^{n - 1} \frac{n - j}{(m + 1) (m + 2)} E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{3}} \\ = \frac{- 16}{n (m + 1) (m + 2)} \\ \times {\sum_{j = n - m}^{n - 2} E {[σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{3} - E {[σ_{n - 1, m} - \frac{2 n}{(m + 1) (m + 2)}]}^{3}} \\ + \frac{16 m}{n (m + 1) (m + 2)} E {[σ_{n - 1 - m, m} - \frac{2 (n - m)}{(m + 1) (m + 2)}]}^{3} \\ = \frac{16}{(m + 1) (m + 2)} \sum_{j = n - m}^{n - 2} E [(\frac{- σ_{j, m}}{n} + \frac{2 (j + 1)}{n (m + 1) (m + 2)}) {(σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)})}^{2}] \\ + \frac{16}{(m + 1) (m + 2)} E [(\frac{- σ_{n - 1, m}}{n} + \frac{2}{(m + 1) (m + 2)}) {(σ_{n - 1, m} - \frac{2 n}{(m + 1) (m + 2)})}^{2}] \\ + \frac{16}{(m + 1) (m + 2)} \\ \times E [(\frac{σ_{n - 1 - m, m}}{n} - \frac{2 (n - m)}{n (m + 1) (m + 2)}) {(σ_{n - 1 - m, m} - \frac{2 (n - m)}{(m + 1) (m + 2)})}^{2}] \\ < \frac{16}{(m + 1) (m + 2)} \sum_{j = n - m}^{n - 2} [0 + \frac{2 (j + 1)}{n (m + 1) (m + 2)}] E {(σ_{j, m} - \frac{2 (j + 1)}{(m + 1) (m + 2)})}^{2} \\ + \frac{32}{(m + 1) (m + 2)} E [(0 + \frac{1}{(m + 1) (m + 2)}) {(σ_{n - 1, m} - \frac{2 n}{(m + 1) (m + 2)})}^{2}] \\ + \frac{16}{(m + 1) (m + 2)} E [(\frac{σ_{n - 1 - m, m}}{n} - 0) {(σ_{n - 1 - m, m} - \frac{2 (n - m)}{(m + 1) (m + 2)})}^{2}] \\ < \frac{32 d_{m}^{2}}{{[(m + 1) (m + 2)]}^{2}} \sum_{j = n - m}^{n - 2} \frac{{(j + 1)}^{2}}{n} + \frac{32 n d_{m}^{2}}{{[(m + 1) (m + 2)]}^{2}} + \frac{16 (n - m) d_{m}^{2}}{(m + 1) (m + 2)} \\ = O (n + 1) . \end{array}

(19)

The last term of (17),

\begin{array}{l} \frac{12 d_{m}^{2}}{n} {\sum_{j = n - m}^{n - 1} {[\frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{2} (j + 1) - \sum_{j = n - 1 - m}^{n - 2} {[\frac{2 (j + 1)}{(m + 1) (m + 2)}]}^{2} (j + 1)} \\ = \frac{12 d_{m}^{2}}{n} {{[\frac{2 n}{(m + 1) (m + 2)}]}^{2} n - {[\frac{2 (n - m)}{(m + 1) (m + 2)}]}^{2} (n - m)} \\ < \frac{48 d_{m}^{2} [n^{3} - {(n - m)}^{3}]}{n {[(m + 1) (m + 2)]}^{2}} = O (n + 1) . \end{array}

(20)

By (17), (18), (19), and (20), we have

E {[σ_{n, m} - \frac{2 (n + 1)}{(m + 1) (m + 2)}]}^{4} < \frac{n + 1}{n} E {[σ_{n - 1, m} - \frac{2 n}{(m + 1) (m + 2)}]}^{4} + O (n + 1) .

(21)

Repeating (21), we have

\frac{E {[σ_{n, m} - \frac{2 (n + 1)}{(m + 1) (m + 2)}]}^{4}}{n + 1} < \dots < \frac{E {[σ_{2 m + 1, m} - \frac{2 (2 m + 2)}{(m + 1) (m + 2)}]}^{4}}{2 m + 2} + O (1) .

(22)

By (22), for n large enough and fixed m, we have

E {[\frac{σ_{n, m}}{n + 1} - \frac{2}{(m + 1) (m + 2)}]}^{4} < O (\frac{1}{n^{3}}) .

(23)

By the Chebyshev inequality, the Borel-Cantelli lemma and (23), for arbitrary

ε > 0

, we have

P (⋃_{n \geq m} {| \frac{σ_{n, m}}{n + 1} - \frac{2}{(m + 1) (m + 2)} | \geq ε}) \to 0 (m \to \infty) .

(24)

By (24), (16) holds. This is the end of the proof. □

Notes

Declarations

Acknowledgements

This work is supported by Anhui Scientific Research Foundation (KJ2012A205, KJ2012B117, KJ2013B162) and Chaohu University Scientific Research Foundation.

Authors’ Affiliations

(1)

Department of Mathematics, Chaohu University, Chaohu, 238000, P.R. China

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