The Subtree Size Profile of Plane-oriented Recursive Trees

Transcription

1 The Subtree Size Profile of Plane-oriente Recursive Trees Michael FUCHS Department of Applie Mathematics National Chiao Tung University Hsinchu, 3, Taiwan Abstract In this extene abstract, we outline how to erive limit theorems for the number of subtrees of size on the fringe of ranom plane-oriente recursive trees. Our proofs are base on the metho of moments, where a complex-analytic approach is use for constant an an elementary approach for which varies with n. Our approach is of some generality an can be applie to other simple classes of increasing trees as well. 1 Introuction Several recent stuies have been concerne with the noe profile of roote ranom trees, where the noe profile is efine as the number of noes at istance from the root; for ranom binary search trees an recursive trees see Chauvin et al. [4], Chauvin et al. [5], Drmota an Hwang [11], [12], Fuchs et al. [22]; for ranom plane-oriente recursive trees see Hwang [23]; for other types of ranom trees see Drmota an Gittenberger [1], Drmota et al. [13], Drmota an Szpanowsi [14], Par et al. [25]. Here, we are going to investigate another in of profile which is efine as the number of subtrees of size. This profile is calle subtree size profile an has so far only been investigate for ranom binary search trees, ranom recursive trees an ranom Catalan trees; see Chang an Fuchs [3], Dennert an Grübel [7], Feng et al. [16], Feng et al. [17], Feng et al. [18], Fuchs [21]. Similar to the noe profile, the subtree size profile is an important tree characteristic carrying a lot of information on the shape of a tree. For instance, total path length (sum of istances of all noes to the root an Wiener inex (sum of istances between all noes can be easily compute from the subtree size profile. Also, as we will explain in more etails below, results about the subtree size profile will in turn entail results about the occurrence of pattern sizes. Stuying patterns in ranom trees is an important issue with many applications in computer science (for instance in Supporte by National Science Council uner the grant NSC M-9-7-MY2. the context of compressing; see Devroye [8] an Flajolet et al. [19] an mathematical biology (see [3] an Rosenberg [26]. In this paper, we will consier the subtree size profile for ranom plane-oriente recursive trees (or ranom PORTs for short. Ranom PORTs have surface in several recent applications sometimes uner ifferent names such as heaporere trees or scale-free ranom trees. They are for instance use as one of the most simplest moel of ranom networs; see Barabási an Albert [1] an the thorough iscussion in [23]. We will start by efining ranom PORTs. First, PORTs of size n are roote plane trees with n noes that are labele such that the sequence of labels from the root to any noe is increasing. Alternatively, PORTs can be efine via a tree evolution process: start with the root; inuctively assume that a noe of out-egree has + 1 free places (before the first chil, between the first an secon chil, etc.; attach the next noe to a free place. Stop when you have attache n noes. It is easy to see that this gives the same sets. Also, from the secon efinition, we can easily count the number τ n of PORTs of size n. Therefore, observe that the sum over all out-egrees in any tree of size i equals i 1. Hence, the number of free places after inserting i noes equals 2i 1. Finally, τ n is the prouct over all free places after inserting i noes with 1 i n 1. Thus, τ n 1 3 (2n n n!, where ( 2n 2 n 1 /n are the (shifte Catalan numbers. Note that the exponential generating function P (z of τ n is given by 1 an satisfies the ifferential equation (1.1 z P (z 1 1 P (z 1 + P (z + P (z2 + with initial conition P (. The latter equation can also be obtaine by symbolic combinatorics: a PORT without the root (this is the left-han-sie is the orere sequence of PORTs tangling from the root (this is the right-han-sie. A ranom PORT of size n is now efine as a PORT which is chosen from all PORTs of size n with probability

2 1/τ n. Again, an alternative efinition can be given via the above tree evolution process: attach the next noe uniformly to one of the free places. Note that uner this tree evolution process, a noe with high out-egree is more liely to attract the next noe. This preferential attachment rule is one of the reasons of the importance of PORTs. In the sequel, we will be intereste in the subtree size profile of ranom PORTs which is a ouble-inexe ranom variable enote by X n,. We will erive limit laws for fixe an for tening to infinity as n tens to infinity. More precisely, we will prove the following result. THEOREM 1.1. (i (Normal range Let n such that 1 o ( n. Then, X n, µ n, σ n, N(, 1, where µ n, (2n 1/(4 2 1 an, as n, ( σn, ( (2 3!! 2 ( 1! n. (2 + 1 (ii (Poisson range Let n such that c n as n. Then, X n, Poisson(2 1 c 2. (iii (Degenerate range Let n such that < n an n o( as n. Then, L X 1 n,. We will explain some consequences of this result for the occurrences of pattern sizes (the following interpretation of our result was pointe out to us by one of the anonymous referees. Therefore, observe that the number of patterns of size is given by C (the number of roote plane trees. From Stirling s formula, we have (1.2 C π 1/2 3/2 4 1 (. Consequently, all patterns can only occur up to a size which is O(log n. Beyon this, certain patterns cease to exist. Our result on the other han shows that espite of this, all pattern sizes o still occur up to o( n. Pattern sizes of orer n then only exist sporaically an are Poisson. Finally, all pattern sizes beyon the orer of n are highly unliely. The non-existence of patterns beyon n is consistent with other well-nown properties of PORTs. For instance, if we enote by T n the total path length, then the orer of T n is nown to be O(n log n. Moreover, as mentione above, T n can easily be compute from X n, as n 1 T n X n,. Hence, there cannot be patterns with sizes much beyon n log n. Our result shoul be compare with the corresponing results for ranom binary search trees an recursive trees in [16] where the same phenomena was observe (with a slightly ifferent expression for the mean an a more simpler expression for the variance. The proof metho from [16], however, cannot be applie here since it reste on an exact expression for all centere moments which oes not seem to be available here. So, we will have to evise a new metho of proof. Our new metho will use an inuctive proceure calle moment pumping that was use in several recent stuies in the analysis of algorithms (see Chern et al. [6] an references therein. In comparison to [16], our metho will incorporate the cancelations from [16] alreay in the inuction step maing the resulting proof much simpler. Another avantage of our metho is that it can be applie to other classes of ranom trees as well. This will be briefly inicate in a final section, where we are going to apply our metho to other simple classes of ranom increasing trees, thereby showing that the above phenomena is universal for these tree families. We mention in passing that the subtree size profile for simple classes of ranom trees (for a efinition see Drmota [9] was investigate as well. For instance, Catalan trees were treate in [3] an a similar result as above was prove with the critical point n moving up to n 2/3. Similar to simple families of ranom increasing trees, the same phenomena is again expecte to hol universally for many other simple families of ranom trees. Before presenting etails, we will give a short setch of the paper. In the next section, we will loo at the case of constant an erive mean value an variance. Therefore, we will use complex-analytic tools, more precisely, singularity analysis with its closure properties. This metho combine with inuction can be extene to erive the first orer asymptotics of all higher centere moments, too. By the Fréchet-Shohat Theorem, this then implies Theorem 1.1, part (i for constant. As for varying, we will use an elementary approach (in the number-theoretic sense, i.e., no complex analysis is use again base on moments an inuction. Finally, in the last section, we will briefly inicate how our tools can be use to erive similar results for other simple classes of ranom increasing trees, too. 2 Constant Subtree Size - An Analytic Approach Here, we will setch the proof of Theorem 1.1, part (i for constant. Therefore, we will wor with moments.

3 The first step will be to erive the mean value. Next, we will shift-the-mean an show that generating functions of centere moments satisfy a recursive relation (see Lemma 2.1 below. This combine with singularity analysis with its closure properties (see Chapter VI in Flajolet an Segewic [2] will allow us to obtain the singularity expansion of the generating function of the variance from properties of the mean. Then, an asymptotic expansion for the variance will follow from the transfer theorems in [2]. Moreover, this proceure can be generalize to all higher centere moments as well. Finally, our result will follow from the Fréchet- Shohat Theorem (see Lemma 1.43 Elliott in [15]. We mention in passing that one coul alternatively try an approach base on the bivariate generating function (2.4 an Hwang s quasi-power theorem (see [2]. Such an approach has the avantage that one oes not have to treat all moments, but is liely to be restricte to constant only. Our approach via moments, however, also wors for varying as we will emonstrate in Section 3. The starting point of our proof is the following (trivial observation: the number of subtrees in a ranom PORT of size n is the sum of the number of subtrees in all subtrees of the root which are again ranom PORTs. This observation translates into the following istributional recurrence (2.3 X n, N i1 X (i I i, (n > with initial conitions X, 1, X n, for n < an X (i n, X n,. Moreover, X n,, X (i n,, (N, I 1, I 2,... are inepenent ranom variables, where N is the out-egree of the root an I 1,..., I N are the sizes of the subtrees of the root. Due to the uniform probability moel, we have for the joint istribution of (N, I 1, I 2,... π n,r,i1,...,i r : P (N r, I 1 i 1,..., I r i r ( n 1 τi1 τ ir, i 1,..., i r τ n where i 1,..., i r 1 an i i r n 1. Since we are intereste in moments of X n,, we efine the bivariate generating function (2.4 P (z, y n 1 Then, (2.3 translates into (2.5 τ n E (exp(x n, y zn n!. z P 1 (z, y 1 P (z, y +(ey 12 1 C z 1 with initial conition P (, y. Note that P (z, P (z for all 1. As (1.1, the above ifferential equation can also be alternatively obtaine by symbolic combinatorics: a PORT without the root is the sequence of PORTs tangling from the root. Moreover, the number of subtrees of size in a PORT is the sum of the number of subtrees of size in the PORTs tangling from the root (this is the first term of the right-hansie in (2.5 except in the case where the PORT itself has size where we have to a one (this is the correction term on the right-han-sie of (2.5. Next, we compute the mean. Therefore, set M (z n 1 τ n E(X n, zn n!. Then, by ifferentiating the above ifferential equation with respect to y an setting y, we obtain the following ifferential equation z M (z M (z + 21 C z 1 with initial conition M (. The solution of this ifferential equation is easily obtaine as (2.6 M (z 21 C z t 1 1 2tt. From this we can erive the mean value. PROPOSITION 2.1. We have, for n >, Proof. From (2.6, we obtain E (X n, 2n E(X n, n!21 C [z n 1 z ] t 1 1 2tt. τ n Next, 1 z t 1 1 2tt 1 1/2 t 1 1 2tt + 1 z 2 B(, 3/2 (2.7 z /2 1/2 l ( 1! (2 + 1!! 1 t 1 1 2tt 1 ( 1 l ( 1 l (1 2t l+1/2 t 1 ( 1 ( l+1 l 2l + 3 (l+1, l

4 where B(x, y is the beta function. Plugging this into the above expression gives for n > E(X n, n!21!c 1 τ n (2 + 1!! [zn ] 2n , where the last line follows by a simple computation. REMARK 2.1. Note that Theorem 1.1, part (iii follows from the above exact expression of the mean value. Next, we consier the variance an higher centere moments. Therefore, we shift-the-mean. Set µ : 2/(4 2 1 an P (z, y n 1 Then, (2.5 becomes τ n E(exp((X n, µny zn n! P (ze µy, y. z P e µy (z, y 1 P (z, y + (ey 1e µy 2 1 C z 1 with initial conition P (, y. Next, set Ā [m] (z τ n E(X n, µn m zn n!. n 1 Then, again by ifferentiation, we obtain z Ā[m] Ā[m] (z (z with initial conition Ā[m] ( an [m] + B (z B [m] (z (( µ + 1 m ( µ m 2 1 C z m 1 i ( m m i i 1 ( µ i y i 1 P (z, y ( m 1 Ā [i1+1] (z i 1+i 2+i 3m 1 i 1<m 1 i2 1 y i2 1 P (z, y where i 1 y i 1 P (z, y i l y i 1, i 2, i 3 ( i µ l l z Ā[l] (z y i3 1 y i3 1 P, (z, y y y ( (1 µ i ( µ i 2 1 C z 1. B [m] Note that (z is a function of Ā[i] (z with i < m. The above ifferential equation is easily solve. LEMMA 2.1. We have, Ā [m] (z 1 z B [m] (t 1 2tt. This is a recurrence for the generating functions of all centere moments. Hence, we can erive singularity expansions of these generating functions by the closure properties in Chapter VI in [2] an inuction. Asymptotic expansions for the centere moments will then follow from the transfer theorems of Chapter VI in [2]. We will first emonstrate this proceure by treating the variance. Since we want to use the results from Chapter VI in [2], we nee to show that the generating functions are analytic in a omain of the form: for R > 1 an < φ < π/2, (R, φ {z : z < R, z 1/2, arg(z 1/2 > φ}. We call such a omain a -omain. Moreover, a function is calle analytic in a -omain if it is analytic in a -omain for some R an φ. Now, we prove the following result. PROPOSITION 2.2. Ā [2] (z is analytic in a -omain. Moreover, we have the singularity expansion where Ā [2] (z σ O(1 (z 1/2, z, σ (2 3!! 2 ( ( 1! ( Proof. First, note that [2] B (z is a function of Ā [1] (z M zµ (z. Hence, from the expression for M (z erive in the proof [2] of Proposition 2.1, we obtain that B (z is -analytic an B [2] (z O(1/( as z 1/2 an z. Consequently, by Lemma 2.1 an Theorem VI.9 in [2], Ā [2] (z is also -analytic an Ā [2] (z as z 1/2 an z, where c 1/2 c + O(1 B [2] (t 1 2tt. Finally, the claime expression for c is obtaine from the [2] exact expression of B (z (by plugging in the exact expression of M (z an a straightforwar computation.

5 Applying the transfer theorems in [2] gives now the following consequence. COROLLARY 2.1. As n, Var (X n, σ 2 n + O ( n 1/2. REMARK 2.2. The above metho can be use to erive longer expansions of the variance, too. As for higher centere moments, we use the same proceure together with inuction (etails will be given in the journal paper. PROPOSITION 2.3. Ā [m] (z is analytic in a -omain for all m 3. Moreover, we have the singularity expansions ( Ā [2m 1] (z O ( 3/2 m (z 1/2, z an Ā [2m] (2m!(2m 3!!σ2m (z 4 m ( 1/2 m m! + O ( ( 1 m (z 1/2, z. Theorem 1.1, part (i for constant follows from this by the transfer theorems in [2] an the Fréchet-Shohat Theorem. 3 Variable Subtree Size - An Elementary Approach Now, we will setch the proof of the remaining cases of Theorem 1.1. Therefore, assume that n with as n. We will concentrate on part (i, part (ii being prove similarly. As in the previous section, our proof is base on moments an inuction. However, here we will wor irectly with the unerlying sequences instea of their generating functions. Moreover, we will use a slightly ifferent starting point. Therefore, observe that a ranom PORT can be ecompose into two ranom PORTs, one being the leftist subtree an the other the remaining tree. This gives the following istribution recurrence (3.8 X n, XIn, + X n I n, 1 {n In} (n > with initial conitions X, 1, X n, for n < an Xn, X n,. Here, I n is the size of the leftist subtree. Due to the uniform ranom moel, we have π n,j : P (I n j 2(n jc j j n (1 j < n. Since the mean value was alreay erive in the previous section, we can immeiately concentrate on centere moments. Therefore, set, if n <, µ n, 1, if n ; 2n/(4 2 1, if n > ; an P n, (z E(exp((X n, µ n, z. Then, from (3.8 P n, (z 1 j<n where P n, (z 1 for n an π n,j Pj, (z P n j, (ze n,j,z (n >, n,j, µ j, + µ n j, µ n, 1 {jn }. Next, set Ā [m] n, E(X n, µ n, m. Differentiating yiels the following recurrence (3.9 Ā [m] n, 1 j<n (Ā[m] π n,j j, + Ā[m] n j, + where Ā[m] n, for n an B [m] n, (3.1 i 1+i 2+i 3m i 1,i 2<m ( m i 1, i 2, i 3 1 j<n The above recurrence can be easily solve. LEMMA 3.1. For n >, (3.11 Ā [m] n, +1 j n [m] B n, (n >, π n,j Ā [i1] j, Ā[i2] n j, i3 n,j,. C j (n + 1 j Proof. This is a straightforwar justification. B[m] j,. [m] Consequently, since B n, is a function of Ā[i] n, with i < m, we again have a recursive relation for Ā[m] n,. Therefore, we can again use inuction to obtain first orer asymptotics of all higher centere moments. However, to estimate error terms, we will nee a uniform boun as well which is erive by another inuction (see [22] where the same approach was use in a ifferent context. We first treat the variance. PROPOSITION 3.1. (i For, n 1, ( n Var(X n, O 2. (ii Let n such that as n. Then, as n, Var(X n, n 2 2. Proof. Here, (3.1 becomes B [2] n, π n,j 2 n,j,. 1 j<n

6 Next, note that (3.12 B [2] n, O ( π n, ( 1 O 2 + (n C. n We will plug this into Lemma 3.1 an treat the resulting two sums separately. First, for the first sum, j n O O C j (n + 1 j+1 j n3/ j n ( n 1 2 /n j 3/2 (n + 1 j 1/2 x 3/2 (1 x 1/2 x ( n O, 5/2 where we have use (1.2. Next, for the secon sum C +1 j n C C 1 j n (j C j (n + 1 j+1 j j +1 j n C j (n + 1 j +1 j C j (n + 1 j+1 j j C (n C C j (n + 1 j+1 j j +1 j n 1 2 E(X n, 2n 1 2(2 + 1(2 1, where the first claim in the last line is prove by showing that the expression solves the recurrence for the mean. This proves part (i. As for part (ii, observe that (3.12 can be refine to B [2] n, π n, 2 n,, + O ( 1/ 2. Now, we again use Lemma 3.1. The error term is treate as above. As for the main term, observe that 2 n,, 1 as n. Consequently, C j (n + 1 j+1 j π j, 2 j,, +1 j n 2C +1 j n (j C j (n + 1 j+1 j j n 2 2, where the last line follows as above. This proves part (ii. The next step is to use inuction to generalize the previous result to all higher centere moments. The proof is long an cumbersome an we will not give etails here. PROPOSITION 3.2. (i For, n 1 an m 2, { Ā [m] n ( n } m/2 n, (max O 2, 2. (ii Let n such that o( n an as n. Then, as n, ( ( Ā [2m 1] n m 1/2 n, o 2 ; ( Ā [2m] n m n, g m, 2 2 for m 1, where g m (2m!/(2 m m!. From this, our claime result follows by the Fréchet- Shohat Theorem. 4 Other Simple Classes of Increasing Trees In this final section, we inicate some extensions of our result to other simple classes of increasing trees. We will be rather brief an postpone further etails to the journal version of this paper. Simple classes of increasing trees were introuce in Bergeron et al. [2] an are efine as follows: first, an increasing tree is a roote, plane, noe-labele tree with the sequence of labels from the root to any noe being increasing. A simple class of increasing trees is then efine as the class of increasing trees together with a weight sequence φ r with φ > an φ r > for some r 2. The orinary generating function of the weight sequence, i.e., φ(ω r φ r ω r is calle weight function. Using the weight sequence, a weight is attache to every increasing tree T as follows ω(t v T φ (v, where the prouct runs over all noes of T an (v is the out-egree of v. Next, we equip a simple class of increasing trees with a probability moel, where the probability of T is proportional to its weight, i.e., if τ n ω(t, T has n noes then the probability of a tree T with n noes is ω(t /τ n. Note that if P (z enotes the exponential generating function of τ n then z P (z φ(p (z φ + φ 1 P (z +

7 with initial conition P (. As (1.1, this ifferential equation can again be erive by symbolic combinatorics: an increasing tree without the root (this is the left-han-sie is the sequence of increasing trees tangling from the root where we have to multiply with φ if the root has out-egree (this is the right-han-sie. It is easy to see that ranom binary search trees, ranom recursive trees an ranom PORTs are all simple classes of increasing trees. Note that all of them can be generate by a tree evolution process. It is interesting to as which other simple classes of increasing trees amit such a construction (via a natural tree evolution process. This question was solve in Panholzer an Proinger [24], where it was shown that up to scaling such a construction exists if an only if the class belongs to following three types. Ranom -ary trees: φ(ω (1 + t, where {2, 3, 4,...}. Ranom recursive trees: φ(ω e t. Generalize ranom PORTs: φ(ω (1 t r+1, where r > 1. Using our previous approach, a similar result as Theorem 1.1 can be prove for those classes as well (for instance, again by symbolic combinatorics, one easily erives a ifferential equation similar to (2.5 for the bivariate generating function. We just state our result for generalize ranom PORTs. First, the mean value of generalize ranom PORTs has again a simple expression. THEOREM 4.1. We have for n > µ n, : E(X n, (r 1(rn 1 (r + r 1(r 1. Moreover, results for the variance an limit laws can be given as well. We just state the result for varying. THEOREM 4.2. (i (Normal range Let n such that as n. Then, where, as n, X n, µ n, σ n, σ 2 n, r 1 r N(, 1, n 2. (ii (Poisson range Let n such that c n as n. Then, X n, Poisson((r 1r 1 c 2. (iii (Degenerate range Let n such that < n an n o( as n. Then, L X 1 n,. Acnowlegments We than Marus Kuba an Alois Panholzer for bringing the current topic to our attention an for some insightful iscussions. Moreover, we than Brigitte Vallée an the anonymous referees for many helpful comments. References [1] A.-L. Barabási an R. Albert (1999. Emergence of scaling in ranom networs, Science, 286, [2] F. Bergeron, P. Flajolet, an B. Salvy (1992. Varieties of increasing trees, Lectures Notes in Computer Science, 581, [3] H. Chang an M. Fuchs (21. Limit theorems for pattern in phylogenetic trees, Journal of Mathematical Biology, 6, [4] B. Chauvin, M. Drmota, an J. Jabbour-Hattab (21. The profile of binary search trees, Annals of Applie Probability, 11, [5] B. Chauvin, T. Klein, J.-F. Marcert, an A. Rouault (25. Martingales an profile of binary search trees, Electronic Journal of Probability, 1, [6] H.-H. Chern, M. Fuchs, an H.-K. Hwang (27. Phase changes in ranom point quatrees, ACM Transactions on Algorithms, 3, 51 pages. [7] F. Dennert an R. Grübel (21. On the subtree size profile of binary search trees, Combinatorics, Probability an Computing, 19, [8] L. Devroye (1998. On the richness of the collection of subtrees in ranom binary search trees, Information Processing Letters, 65, [9] M. Drmota. Ranom Trees. SpringerWienNewYor, 29. [1] M. Drmota an B. Gittenberger (1997. On the profile of ranom trees, Ranom Structures an Algorithms, 1, [11] M. Drmota an H.-K. Hwang (25. Bimoality an phase transitions in the profile variance of ranom binary search trees, SIAM Journal on Discrete Mathematics, 19, [12] M. Drmota an H.-K. Hwang (25. Profile of ranom trees: correlation an with of ranom recursive trees an binary search trees, Avances in Applie Probability, 37, [13] M. Drmota, S. Janson, an R. Neininger (28. A functional limit theorem for the profile of search trees, The Annals of Applie Probability, 18, [14] M. Drmota an W. Szpanowsi (21. The expecte behavior of igital search tree profile, submitte. [15] P. D. T. A. Elliott. Probabilistic Number Theory I. Central Limit Theorems. Springer, New-Yor, [16] Q. Feng, H. Mahmou, an A. Panholzer (28. Phase changes in subtree varieties in ranom recursive trees an binary search trees, SIAM Journal on Discrete Mathematics, 22, [17] Q. Feng, H. Mahmou, an C. Su (27. On the variety of subtrees in a ranom recursive tree, Technical report, The George Washington University, Washington, DC. [18] Q. Feng, B. Miao, an C. Su (26. On the subtrees of

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