The Subtree Size Profile of Bucket Recursive Trees

Transcription

1 Iranian Journal of Mathematical Sciences and Informatics Vol., No. (206, pp - DOI: /ijmsi The Subtree Size Profile of Bucket Recursive Trees Ramin Kazemi Department of Statistics, Imam Khomeini International University, Qazvin, Iran. r.kazemi@sci.ikiu.ac.ir Abstract. Kazemi (204 introduced a new version of bucket recursive trees as another generalization of recursive trees where buckets have variable capacities. In this paper, we get the p-th factorial moments of the random variable S n, which counts the number of subtrees size- profile (leaves and shows a phase change of this random variable. These can be obtained by solving a first order partial differential equation for the generating function correspond to this quantity. Keywords: Bucket recursive tree, Subtree size profile, Factorial moments Mathematics subject classification: 05C05, 60F05.. Introduction Trees are defined as connected graphs without cycles, and their properties are basics of graph theory. For example, a connected graph is a tree, if and only if the number of edges equals the number of nodes minus. Furthermore, each pair of nodes is connected by a unique path []. A rooted tree is a tree with a countable number of nodes, in which a particular node is distinguished from the others and called the root node [32]. Thenode profile isdefinedasthenumberofnodesatdistancek fromtheroot in a tree. Several studies have been concerned on this quantity; for random binary search trees and recursive trees see [4, 5, 9, 0] and [9]; for random Received 9 January 203; Accepted 02 March 205 c 206 Academic Center for Education, Culture and Research TMU

2 2 R. Kazemi plane-oriented recursive trees see [20]; for other types of random trees see [8,, 2, 30] and [26]. There is another kind of profile which is defined as the number of subtrees of size k. This kind is called subtree size profile and has been investigated for random binary search trees, random recursive trees and random Catalan trees; see [3, 6, 3, 4, 5] and [8]. This kind of profile is an important tree characteristic carrying a lot of information on the shape of a tree. For instance, total path length (sum of distances of all nodes to the root and Wiener index (sum of distances between all nodes can be easily computed from the subtree size profile [7]. Also, studying patterns in random trees is an important issue with many applications in computer science (see [7] and [6] and mathematical biology (see [3] and [3]. Meir and Moon [28] defined recursive trees as the variety of non-plane increasing trees [2] such that all node degrees are allowed. In this model, the capacity of nodes is [2]. Mahmoud and Smythe [29] introduced bucket recursive trees as a generalization of random recursive trees where the capacity of buckets is fixed. In this paper, we will consider another bucket recursive trees, i.e., bucket recursive trees with variable capacities of buckets that introduced by Kazemi (204. He studied the following random variables in this model: the depth of the largest label [23], the first Zagreb index [22], the eccentric connectivity index [24] and the branches [25]. Also, Kazemi and Haji showed a phase change in the distribution in these models [27]. Our results for b = reduce to the previous results for random recursive trees [3, 4]. We define the tree below for the reader s convenience [23]. Definition.. A size-n bucket recursive tree T n with variable bucket capacities and maximal bucket size b starts with the root labeled by. The tree grows by progressive attraction of increasing integer labels: when inserting label j + into an existing bucket recursive tree T j, except the labels in the non-leaf nodes with capacity < b all labels in the tree (containing label compete to attract the label j +. For the root node and nodes with capacity b, we always produce a new node j +. But for a leaf with capacity c < b, either the label j + is attached to this leaf as a new bucket containing only the label j + or is added to that leaf and make a node with capacity c +. This process ends with inserting the label n (i.e., the largest label in the tree. By definition, a node v with capacity c(v < b has the out-degree 0 or. In Figure, we diagrammatically show the step-by-step growth of a tree of size with b = 2. We consider the random variable S n,k, which counts the number of buckets that are the root of a subtree of T n with size k. More precisely, we study the subtree size profile S n, in our model (=leaves.

3 The Subtree Size Profile of Bucket Recursive Trees 3 Figure. The step-by-step growth of a tree of size n = with maximal bucket size b = Partial Differential Equation AclassT ofafamilyofbucket-increasingtreescanbedefinedinthefollowing way (see [23, Section 2] for details. A sequence of non-negative numbers (α k k 0 with α 0 > 0 and a sequence of non-negative numbers β,β 2,,β b are used to define the weight w(t of any ordered tree T by w(t = Π v w(v, where { αd(v, v is the root or c(v = b w(v = (2. β c(v, c(v < b and d(v denotes the out-degree of node v. Let L(T be the set of different increasing labelings of the tree T with distinct integers {,2,..., T } (. denotes the size of sets. Then the family T consists of all trees T together with their weights w(t and the set of increasing labelings L(T. We define the exponential generating function T n,b (z = n= T n,b z n n!, (2.2

4 4 R. Kazemi where T n,b := T =nw(t L(T is the total weights and L(T := L(T. If r is the out-degree of the root node, then r T n,b = (n!(b!n( P k i, n,b, (2.3 b where P ki is the set of all trees of size k i and T n,b (0 = 0 [23]. Let S k (z,u be the moment generating function S k (z,u = z n P(S n,k = mt n,b n! um n m 0 = r P(S n,k = m (b!n( P k i z n b n um. (2.4 n m 0 According to the definition of the tree, the probabilities of P(S n,k = m satisfy (n i, m i 0 and n > k P(S n,k = m = n T n,b T n r,b r! n r n,...,n r T n,b + +n r=n P(S n,k = m P(S nr,k = m r, (2.5 m + +m r=m with initial values P(S k,k = =, P(S n,k = 0 = for n < k where Tn i,b is the total weights of the ith subtree. Thus recurrence (2.5 leads to the following functional equation [23] z S k(z,u = b! ( r P k i e Sk(z,u +(u z k, (k (2.6 with initial condition S k (0,u = 0. For b =, i.e., random recursive trees, Feng, et al. obtained a limit theorem for the subtree size profile by considering both k fixed and k = k(n dependent on n. Using analytic methods they characterized for the tree the phase change behavior of S n,k [4]. For b >, there is no unique solution of (2.6 for all k. Suppose β(r,b = b! r P k i. Then S (z,u = log ( e z(u β(r,b( +. (2.7 u u 3. Preliminaries We can rewrite S (z,u as follows: S (z,u = log z +(u zβ(r,b. (3. 0 e(u β(r,bt dt

5 The Subtree Size Profile of Bucket Recursive Trees 5 Set y = u and = b (b! n( r P k i. Thus S (z,+y = P(S n, = m zn n (+ym n m 0 = m m p P(S n, = m zn y p n p!, n m 0 p=0 where m p = m(m (m p+. Hence E(S p n, = np![z n y p ]S (z,+y, where [z n ]f(z denote the operation of extracting the coefficient of z n in the formal power series f(z = f n z n. Also log z 0 eyβ(r,bt dt ( For p, [y p ]log = log z 0 = log z z j = log z +log = [y p ] i p (zβ(r,b p = i = p! (tyβ(r,b j j 0 j! dt z y j (zβ(r,b j z j (j+! i z i z j p (zβ(r,b p i y j (zβ(r,b j (j+! z y j (zβ(r,b j z j (j+!. y j (zβ(r,b j (j +! ( z z i j + +j i=p,j q i ( z z j + +j i=p,j q i i k= (j k +!. p i k= (j. k + j,...,j i Set I(A := {, if A is true 0, otherwise.

6 6 R. Kazemi From (3., [y p ]S (z,+y = p! p (zβ(r,b p i j + +j i=p,j q i z z i k= (j k + ( p j,...,j i + zβ(r,bi(p =. ( Main Results Theorem 4.. Let β(r,b = b! r P k i and = b (b! n( r P k i. Then E(S n, = { (b!, n = n 2 β(r,b, n 2 (4. and for p 2, p E(S p n, = β(r,b p ( n p n i i ( p i k= (j k + j,...,j i j + +j i=p,j q I(n p+. (4.2 Let s(m,n be the mth Stirling number of order n (of the second kind. Then in view of the classical relation p E(S p n, = s(p,ie(s i n,. Thus we can get closed formulas for ordinary p-th moments. We use the notations D and P to denote convergence in distribution and in probability, respectively. The standard random variable Poi(λ and N(µ,σ 2 appear in the following theorem for the Poisson distributed with parameter λ > 0 and the normal distributed with mean µ and variance σ 2, respectively. These random variables appear in the results as limiting random variables. Theorem 4.2. Let S n, be the subtree size- profile in size-n bucket recursive trees with variable capacities of buckets. Then i P S n, 0, as nβ(r,b. ii D S n, Poi c 2, as nβ(r,b c > 0,

7 The Subtree Size Profile of Bucket Recursive Trees 7 and otherwise no limiting distribution exists for S n,. iii S n, n β(r,b 2 6nβ(r,b 5nβ(r,b 2 2 D N(0,, as 5. Proofs nβ(r,b 0. Proof of Theorem 4.. The formula (3.2 immediately gives For p 2, ( E(S n, = n [z n ] zβ(r,b+ β(r,b z 2 2 z ( = n β(r,bi(n = + β(r,b I(n 2 2 = n β(r,b I(n = + n β(r, b I(n 2. 2 E(S p n, = np![z n y p ]S (z,+y p = n [z n β(r,b p z p+i ] i ( z i p i j + +j i=p,j q k= (j k + j,...,j i p = n β(r,b p n p I(n p+ i i p i k= (j. k + j,...,j i Corollary 5.. For p = 2, and j + +j i=p,j q E(S 2 n, = E(S n,(s n, = n β(r,b2 For b =, {, n = E(S n, = n 2, n 2 E(S 2 n, = n 3 + n(n 3, n 3 4 ( 3 + n 3, n 3. 4 that are the same results for the (ordinary recursive trees [4].

8 8 R. Kazemi Proof of Theorem 4.2. Suppose nβ(r,b λ > 0. Thus from Theorem 4., E(S n, λ 2. It is obvious that p = p![z p ](e z i p![z p ]e iz = i p. j,...,j i j + +j i=p,j q i For p 2, p E(S p n, = β(r,b p n p n i i p i k= (j k + j,...,j i j + +j i=p,j q p n β(r,b p p β(r,b p n i (i! ip p nβ(r,b p i=0 p nβ(r,b = p exp i n i i p i! i nβ(r,b i! ( nβ(r,b since p i. By assumption nβ(r,b 0. Then for all p, E(Sp n, 0. i.e., the random variable S n, convergent to a degenerate distribution at point 0. ii From Theorem 4., E(S p n, = A+B, where A = and p nβ(r,b p i n p i j + +j i=p,j q, p i k= (j, k + j,...,j i B = nβ(r,bp n p p! p p 2 p p ( nβ(r,b = +O n. 2 With the same technic of Part (i, A p p β(r, b Thus nβ(r, b exp ( β(r, b A = O nβ(r,b. = O n,

9 The Subtree Size Profile of Bucket Recursive Trees 9 since nβ(r,b λ. Finally for every p, p E(S p nβ(r,b n, = +O 2 Now, if we use the substitution c = ( n 2 λ, then nβ(r,b p λ. 2 c and this proves the Part (ii. iii Let S n, = S n, E(S n, and S (z,s = n E(eSn, zn n. Then S (z,s = n e E(Sn,s E(e Sn,s zn n. From (2.4 and initial conditions of (2.5, By (4., S (e sβ(r,b 2 z,e s = e sβ(r,b 2 ze s + (e sβ(r,b n 2 z P(S n, = m e sm. n n 2m 0 S (z,s = e E(S,s E(e S,s z + n 2 e E(Sn,s E(e Sn,s zn n = S (e sβ(r,b 2 z,e s + (e ( (b! s e sβ(r,b +s 2 z. Now by (3. and just similar to [3] proof is completed. Corollary 5.2. The Theorem 4.2 for b = reduce to the previous results for random recursive trees [3]. Acknowledgments The author would like to expresses its sincere thanks to the referees for their valuable suggestions and comments. References. A. R. Ashrafi, S. Yousefi, A note on the equiseparable trees, Iranian Journal of Mathematical Sciences and Informatics, 2(, (2007, F. Bergeron, P. Flajolet, B. Salvy, Varieties of increasing trees, Lecture Notes in Computer Science, 58, (992, H. Chang, M. Fuchs, Limit theorems for pattern in phylogenetic trees, Journal of Mathematical Biology, 60, (200, B. Chauvin, M. Drmota, J. Jabbour-Hattab, The profile of binary search trees, Annals of Applied Probability,, (200, B. Chauvin, T. Klein, J. F. Marckert, A. Rouault, Martingales and profile of binary search trees, Electronic Journal of Probability, 0, (2005,

10 0 R. Kazemi 6. F. Dennert, R. Grubel, On the subtree size profile of binary search trees, Combinatorics, Probability and Computing, 9, (200, L. Devroye, On the richness of the collection of subtrees in random binary search trees, Information Processing Letters, 65, (998, M. Drmota, B. Gittenberger, On the Profile of random trees, Random Structures and Algorithms, 0, (997, M. Drmota, H. K. Hwang, Bimodality and phase transitions in the profile variance of random binary search trees, SIAM Journal on Discrete Mathematics, 9, (2005, M. Drmota, H. K. Hwang, Profile of random trees: correlation and width of random recursive trees and binary search trees, Advances in Applied Probability, 37, (2005, M. Drmota, S. Janson, R. A. Neininger, Functional limit theorem for the profile of search trees, The Annals of Applied Probability, 8, (2008, M. Drmota, W. Szpankowski, The expected profile of digital search trees, Journal of Combinatorial Theory, Series A, 8, (20, Q. Feng, H. Mahmoud, A. Panholzer, Phase changes in subtree varieties in random recursive trees and binary search trees, SIAM Journal on Discrete Mathematics, 22, (2008, Q. Feng, H. Mahmoud, C. Su, On the variety of Subtrees in a Random Recursive Tree, Technical Report the George Washington University, Washington, DC., Q. Feng, B. Miao, C. Su, On the subtrees of binary search trees, Chinese Journal of Applied Probability and Statistics, 22, (2006, P. Flajolet, X. Gourdon, C. Martinez, Patterns in random binary search trees, Random Structures and Algorithms,, (997, M. Fuchs, The subtree size profile of plane-oriented recursive trees, SIAM, (20, M. Fuchs, Subtree sizes in recursive trees and binary search trees: Berry-Esseen bounds and poisson approximations, Combinatorics, Probability and Computing, 7, (2008, M. Fuchs, H. K. Hwang, R. Neininger, Profiles of random trees: limit theorems for random recursive trees and binary search trees, Algorithmica, 46, (2006, H. K. Hwang, Profiles of random trees: plane-oriented recursive trees, Random Structures and Algorithms, 30, (2007, R. Kazemi, Note on the almost surely convergence in random recursive trees, Far East Journal of Applied Mathematics, 24(, (2006, R. Kazemi, Probabilistic analysis of the first Zagreb index, Transactions on Combinatorics, 2(2, (203, R. Kazemi, Depth in bucket recursive trees with variable capacities of buckets, Acta Mathematics Sinica, English Series, 30(2, (204, R. Kazemi, The eccentric connectivity index of bucket recursive trees, Iranian Journal of Mathematical Chemistry, 5(2, (204, R. Kazemi, Branches in bucket recursive trees with variable Capacities of buckets, UPB Scientific Bulletin, Series A, 77(, (205, R. Kazemi, M. Q. Vahidi-Asl, The variance of the profile in digital search trees, Disctere Mathematics and Theoritical Computer Science, 3(3, (20, R. Kazemi, H. Haji, A phase change in the distribution for bucket recursive trees with variable capacities of buckets, Technical Journal of Engineering and Applied Sciences, 3(4, (203, A. Meir, J. W. Moon, Recursive trees with no nodes of out-degree one, Congressus Numerantium, 66, (988,

11 The Subtree Size Profile of Bucket Recursive Trees 29. H. Mahmoud, R. Smythe, Probabilistic analysis of bucket recursive trees, Theoretical Computer Science, 44, (995, G. Park, H. K. Hwang, P. Nicodeme, W. Szpankowski, Profiles of tries, SIAM Journal on Computing, 38, (2009, N. A. Rosenberg, The mean and variance of the number of r-pronged nodes and r- caterpillars in Yule-generated genealogical trees, Annals of Combinatorics, 0, (2006, H. Yousefi-Azari, A. Goodarzi, Placement of rooted trees, Iranian Journal of Mathematical Sciences and Informatics, (2, (2006,