LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES

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1 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES JAMES ALLEN FILL AND NEVIN KAPUR Abstract. Additive tree functionals represent the cost of many divide-andconquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a n α when α > 0 and (b log n (the so-called shape functional on uniformly distributed binary search trees, sometimes called Catalan trees. The Gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model. Our results give rise to an apparently new family of distributions containing the Airy distribution (α = and the normal distribution [case (b, and case (a as α 0]. The main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their Hadamard product, and the method of moments.. Introduction Binary search trees are fundamental data structures in computer science, with primary application in searching and sorting. For bacground we refer the reader to Chapter of the excellent boo [7]. In this article we consider additive functionals defined on uniformly distributed binary search trees (sometimes called Catalan trees induced by two types of toll sequences [(n α and (log n]. (See the simple Definition.. Our main results, Theorems 3. and 4., establish the limiting distribution for these induced functionals. A competing model of randomness for binary search trees is the random permutation model (RPM; see Section.3 of [7]. While there has been much study of additive functionals under the RPM (see, for example, [7, Section 3.3] and [0, 5,, 3], little attention has been paid to the distribution of functionals defined on search trees under the uniform (Catalan model of randomness. Fill [5] argued that the functional corresponding to the toll sequence (log n serves as a crude measure of the shape of a binary search tree, and explained how this functional arises in connection with the move-to-root self-organiing scheme for dynamic maintenance of binary search trees. He derived a central limit theorem under the RPM, but obtained only asymptotic information about the mean and variance under the Catalan model. (The latter results were rederived in the extension [8] from binary search trees to simply generated rooted trees. In this paper (Theorem 4. we show that there is again asymptotic normality under the Catalan model. Date: June 4, Mathematics Subject Classification. Primary: 68W40; Secondary: 60F05, 60C05. Key words and phrases. Catalan trees, additive functionals, limiting distributions, divide-andconquer, shape functional, Hadamard product of functions, singularity analysis, Airy distribution, generalied polylogarithm, method of moments, central limit theorem. Research supported by NSF grant DMS and DMS-00467, and by The Johns Hopins University s Acheson J. Duncan Fund for the Advancement of Research in Statistics.

2 JAMES ALLEN FILL AND NEVIN KAPUR In [0, Prop. ] Flajolet and Steyaert gave order-of-growth information about the mean of functionals induced by tolls of the form n α. (The motivation is to build a repertoire of tolls from which the behavior of more complicated tolls can be deduced by combining elements from the repertoire. The corresponding results under the random permutation model were derived by Neininger [9]. Taács established the limiting (Airy distribution of path length in Catalan trees [,, 3], which is the additive functional for the toll n. The additive functional for the toll n arises in the study of the Wiener index of the tree and has been analyed by Janson [3]. In this paper (Theorem 3. we obtain the limiting distribution for Catalan trees for toll n α for any α > 0. The family of limiting distributions appears to be new. In most cases we have a description of the distribution only in terms of its moments, although other descriptions in terms of Brownian excursion, as for the Airy distribution and the limiting distribution for the Wiener index, may be possible. This is currently under investigation by the authors in collaboration with others. The uniform model on binary search trees has been also been used recently by Janson [4] in the analysis of an algorithm of Koda and Rusey [6] for listing ideals in a forest poset. This paper serves as the first example of the application of recent results relating the asymptotics of generating functions of sequences to those of their Hadamard product [6] to obtain limiting distributions. First moments for our problems were treated in [6] and a setch of the technique we employ was presented there. (Our approach to obtaining asymptotics of Hadamard products of generating functions differs only marginally from the Zigag Algorithm as presented in [6]. As will be evident soon, Hadamard products occur naturally when one is analying moments of additive tree functionals. The program we carry out allows a fairly mechanical derivation of the asymptotics of moments of each order, thereby facilitating application of the method of moments. Indeed, preliminary investigations suggest that the techniques we develop are liewise applicable to the wider class of simply generated trees; this is wor in progress. The organiation of this paper is as follows. Section establishes notation and states certain preliminaries that will be used in the subsequent proofs. In Section 3 we consider the toll sequence (n α for general α > 0. In Section 3. we compute the asymptotics of the mean of the corresponding additive functional. In Section 3. the analysis diverges slightly as the nature of asymptotics of the higher moments differs depending on the value of α. Section 3.3 employs singularity analysis [9] to derive the asymptotics of moments of each order. In Section 3.4 we use the results of Section 3.3 and the method of moments to derive the limiting distribution of the additive tree functional. In Section 4 we employ the approach again to obtain a normal limit theorem for the shape functional. Finally, in Section 5, we present heuristic arguments that may lead to the identification of toll sequences giving rise to a normal limit.. Notation and Preliminaries We first establish some notation. Let T be a binary search tree. We use T to denote the number of eys (which equals the number of nodes in T. Let L(T and R(T denote, respectively, the left and right subtrees rooted at the children of the root of T.

3 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 3 Definition.. A functional f on binary search trees is called an additive tree functional if it satisfies the recurrence f(t = f(l(t + f(r(t + b T, for any tree T with T. Here (b n n is a given sequence, henceforth called the toll function. We analye functionals of additive type defined on binary search trees uniformly distributed over {T : T = n} for given n. Let X n be such an additive functional induced by the toll sequence (b n. It is well nown that the number of binary search trees on n nodes is counted by the nth Catalan number with generating function CAT( := β n := n + n=0 ( n n, β n n = ( 4. In our subsequent analysis we will mae use of the identity (. CAT ( = CAT(. The mean of the cost function a n := EX n can be obtained recursively by conditioning on the ey stored at the root as n β j β n j a n = (a j + a n j + b n, n. β n This recurrence can be rewritten as n (. (β n a n = (β j a j β n j + (β n b n, n. Recall that the Hadamard product of two power series F and G, denoted by F( G(, is the power series defined by (F G( F( G( := n f n g n n, F( = n f n n and G( = n g n n. Multiplying (. by n /4 n and summing over n we get (.3 A( CAT(/4 = B( CAT(/4 + a 0 b 0, A( and B( are the ordinary generating functions of (a n and (b n respectively. In our analysis we will assume a 0 = b 0 so that the second term in (.3 does not contribute. (Our results can be easily adjusted when this is not the case. We will use (.3 (with a 0 = b 0 as the starting point for our analysis. In the sequel the notation [ ] is used both for Iverson s convention [5,..3(6] and for the coefficient of certain terms in the succeeding expression. The interpretation will be clear from the context. For example, [α > 0] has the value when

4 4 JAMES ALLEN FILL AND NEVIN KAPUR α > 0 and the value 0 otherwise. In contrast, [ n ]F( denotes the coefficient of n in the series expansion of F(. Recall the definition of the generalied polylogarithm: Definition.. For α an arbitrary complex number and r a nonnegative integer, the generalied polylogarithm function Li α,r is defined for < by (log n r Li α,r ( := n α n. The ey property of the generalied polylogarithm that we will employ is n= Li α,r Li β,s = Li α+β,r+s. We will also mae extensive use of the following consequences of the singular expansion of the generalied polylogarithm. Note that neither this lemma nor the ones following mae any claims about uniformity in α or r. Throughout, we use the notation L( := log( Li 0, (. Lemma.3. For real α < and r a nonnegative integer, and any ɛ > 0, as in the sector π + ɛ < arg( < π ɛ, r Li α,r ( = λ (α,r ( α L r ( + O( α ɛ + ( r ζ (r (α[α > 0], λ (α,r =0 ( r Γ ( ( α and ɛ > 0 is arbitrarily small. Proof. By Theorem in [7], (.4 Li α,0 ( Γ( αt α + ( j ζ(α jt j ( l, t = log =, j! l j 0 and for r a positive integer, Li α,r ( = ( r r α r Li α,0(. Moreover, the singular expansion for Li α,r is obtained by performing the indicated differentiation of (.4 term-by-term. To establish the claim we set f = Γ( α and g = t α in the general formula for the rth derivative of a product: r ( r (fg (r = f ( g (r to first obtain ( r r α r [Γ( αtα ] = ( r The claim then follows easily. =0 r =0 l= ( r ( Γ ( ( αt α (log t r The following inverse of Lemma.3 is very useful for computing with Hadamard products. Lemma.4. For real α < and r a nonnegative integer, r ( α L r ( = µ (α,r Li α,r ( + O( α ɛ + c r (α[α > 0], =0

5 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 5 µ (α,r 0 = /Γ( α, c r (α is a constant, and ɛ > 0 is arbitrarily small. Proof. We use induction on r. For r = 0 we have Li α,0 ( = Γ( α( α + O( α ɛ + ζ(α[α > 0] and the claim is verified with µ (α,0 0 = Γ( α and c 0 (α = ζ(α Γ( α. Let r. Then using Lemma.3 and the induction hypothesis we get Li α,r ( = Γ( α( α L r ( [ r r + = λ (α,r l=0 µ (α,r l Li α,r l ( + O( α ɛ + c r (α[α > 0] + O( α ɛ + ( r ζ (r (α[α > 0] r = Γ( α( α L r ( + + O( α ɛ + ( r = = r λ (α,r s=0 µ (α,r r s Li α,s( λ (α,r c r (α + ( r ζ (r (α r = Γ( α( α L r ( + ν s (α,r Li α,s ( s=0 + O( α ɛ + γ r (α[α > 0],, for 0 s r, and Setting and µ (α,r 0 = the result follows. γ r (α := r s ν s (α,r := λ (α,r µ (α,r r s, r = = Γ( α, µ(α,r λ (α,r c r (α + ( r ζ (r (α. = ν(α,r r c r (α = γ r(α Γ( α, [α > 0] Γ( α, r, For the calculation of the mean, the following refinement of a special case of Lemma.3 is required. It is a simple consequence of Theorem of [7]. Lemma.5. When α < 0, Li α,0 ( = Γ( α( α Γ( α α ( α +O( α+ +ζ(α[α > ]. For sae of completeness, we state a result of particular relevance from [6]. ]

6 6 JAMES ALLEN FILL AND NEVIN KAPUR Theorem.6. If f and g are amenable to singularity analysis and f( = O( a and g( = O( b as, then f g is also amenable to singularity analysis. Furthermore (a If a + b + < 0 then f( g( = O( a+b+. (b If < a + b + < + for some integer <, then f( g( = ( j (f g (j (( j + O( a+b+. j! j=0 (c If a + b + is a nonnegative integer then f( g( = a+b j=0 ( j (f g (j (( j + O( a+b+ L(. j! 3. The toll sequence (n α In this section we consider additive functionals when the toll function b n is n α with α > Asympotics of the mean. Since b n = n α, by definition B = Li α,0. Thus, by Lemma.5, B( = Γ(+α( α Γ(+α α + ( α +O( α+ +ζ( α[α < ]. We will now use (.3 to obtain the asymptotics of the mean. First we treat the case α < /. Since CAT(/4 = + O( / as in a suitable indented crown, we have, by part (b of Theorem.6, B( CAT(/4 = C 0 + O( α+, C 0 := B( CAT(/4 = n α β n = 4 n. We now already now the constant term in the singular expansion of B( CAT(/4 at = and henceforth we need only compute lower-order terms. The constant c is used in the sequel to denote an unspecified (possibly 0 constant, possibly different at each appearance. Let s write B( = L ( + R (, and CAT(/4 = L ( + R (, n= L ( := Γ( + α( α Γ( + α α + ( α + ζ( α, R ( := B( L ( = O( α, L ( := ( ( /, R ( := CAT(/4 L ( = O(.

7 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 7 We will analye each of the four Hadamard products separately. First, L ( L ( = Γ( + α [ ( α ( /] By Theorem 4. of [6], and ( α ( / = c + + Γ( + α α + [ ( α ( /] + c. Γ(α ( α+ + O(, Γ(α + Γ( / ( α ( / = c + O( by another application of part (b of Theorem.6, this time with =. Hence L ( L ( = [ L ( L ( ] = + Γ(α ( α+ + O(. π The other three Hadamard products are easily handled as L ( R ( = [ L ( R ( ] = + O( α+, L ( R ( = [ L ( R ( ] = + O(, R ( R ( = [ R ( R ( ] = + O(. Putting everything together, we get B( CAT(/4 = C 0 + Γ(α π ( α+ + O( α+. Using this in (.3, we get (3. A( CAT(/4 = C 0 ( / + Γ(α π ( α + O( α+. To treat the case α / we mae use of the estimate (3. ( / = a consequence of Theorem of [7], so that Γ( / [Li 3/,0( ζ(3/] + O(, B( ( / = Li α,0 ( ( / = Γ( / Li 3 + R(, α,0( c + O( α / α < (3.3 R( = O( L( α = O( α α >. Hence B( CAT(/4 = Γ( / Li 3 + R(, α,0( R, lie R, satisfies (3.3 (with a possibly different c. When α = /, this gives us B( CAT(/4 = Γ( / L( + c + O( /,

8 8 JAMES ALLEN FILL AND NEVIN KAPUR so that (3.4 A( CAT(/4 = π ( / L( + c( / + O(. For α > / another singular expansion leads to the conclusion that (3.5 A( CAT(/4 = Γ(α ( α + R(, π O( / < α < R( = O( L( α = O( α+ α >. We defer deriving the asymptotics of a n until Section Higher moments. We will analye separately the cases 0 < α < /, α = /, and α > /. The reason for this will become evident soon; though the technique used to derive the asymptotics is induction in each case, the induction hypothesis is different for each of these cases Small toll functions (0 < α < /. We start by restricting ourselves to tolls of the form n α 0 < α < /. In this case we observe that by singularity analysis applied to (3., so a n β n 4 n = C 0 π n / + O(n 3/ + O(n α = C 0 π n / + O(n α, a n = n 3 [ + O(n ][C 0 n + O(n α ] = C 0 n + O(n α+ = (C 0 + o((n +. The lead order of the mean a n = EX n does not depend on α. We perform an approximate centering to get to the dependence on α. Define X n := X n C 0 (n +, with X 0 := 0; µ n ( := E X n, with µ n (0 = for all n 0; and ˆµ n ( := β n µ n (/4 n. Let M ( denote the ordinary generating function of ˆµ n ( in the argument n. By an argument similar to the one that led to (., we get, for, ˆr n ( := 4 ˆµ n ( = = 4 n n + + 3=, < β n j 4 n j ˆµ j ( + ˆr n (, n, + + 3=, < ( (,, 3,, 3 b 3 n ˆµ j ( ˆµ n j ( b 3 n n ˆµ j ( ˆµ n j (, for n and ˆr 0 ( := ˆµ 0 ( = µ 0 ( = ( C 0. Let R ( denote the ordinary generating function of ˆr n ( in the argument n. Then, mimicing (.3, (3.6 M ( = R (

9 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 9 with (3.7 R ( = ( C =, < for a nonnegative integer (,, 3 B( := B( B(. }{{} (B( 3 [ 4 M ( M ( ], Note that M 0 ( = CAT(/4. We analye separately the cases 0 < α /4 and /4 < α < /. The proof technique in either case is induction, though the latter case is slightly simpler. Proposition 3.. In the notation introduced above, when 0 < α /4, for, M ( = C ( (α+ + + O( (α+ + +(α ɛ, as in the sector π + ɛ < arg( < π ɛ, ɛ > 0 and ɛ > 0 are arbitrarily small. The C s here are defined by the recurrence (3.8 C = ( Γ(α + C j C j + C 4 j Γ(( α +, ; C = Γ(α. π Proof. For = the claim is true as shown in (3. with C as defined in (3.8. We will now analye each term in (3.7 for. For notational convenience, define α := α +. Also, observe that B( = Li α,0 ( = Γ( + α( α + O( α ɛ by Lemma.3. For this paragraph, consider the case that and are both nonero. It follows from the induction hypothesis that 4 M ( M ( = 4 ( ( [ C ( α + + O( α + +(α ɛ ] [ C ( α + + O( α + +(α ɛ ] = 4 C C ( (+α + + O( (+α ++(α ɛ. If 3 = 0 then the corresponding contribution to R ( is ( C C ( α + + O( α ++(α ɛ. 4 If 3 0 we use Lemma.4 to express 4 M C C ( M ( = 4Γ(( + α Li ( + α +,0( + O( (+α ++(α ɛ C C [( + α < ] ζ( ( + α + 4 Γ(( + α. The corresponding contribution to R ( is then (,, 3 times: C C 4Γ(( + α Li α + 3 +,0(+Li 3α,0( O( (+α ++(α ɛ.

10 0 JAMES ALLEN FILL AND NEVIN KAPUR Now 3 so α <. Hence the contribution when 3 0 is O( α = O( α + 3 = O( α ++(α ɛ. Next we consider the case when is nonero but = 0. In this case using the induction hypothesis we see that 4 M ( M ( = 4 CAT(/4 M ( ( / [ = C ( α + ] + O( α + +(α ɛ = C ( + α + O( α + +(α ɛ. Applying Lemma.4 to the last expression we get 4 M C ( M ( = Γ( α Li α + 3,0( + O( α + +(α ɛ C [ α < ]ζ( α + 3 Γ( α. The contribution to R ( is hence ( times: C Γ( α Li α ,0( + Li 3α,0( O( α + +(α ɛ. Using the fact that α > 0 and 3, we conclude that α < so that, by Lemma.3 and part a of Theorem.6, the contribution is O( α = O( α + 3 the displayed equality holds unless 3 =. When 3 = we get a corresponding contribution to R ( of ( times: C Γ(α Γ(( α α + + O( α ++(α ɛ, ( since for we have α > + (α ɛ. The introduction of ɛ handles the case when α = + α, which would have otherwise, according to part c of Thoerem.6, introduced a logarithmic remainder. In either case the remainder is O( α ++(α ɛ. The case when is nonero but = 0 is handled similarly by exchanging the roles of and. The final contribution comes from the single term both and are ero. In this case the contribution to R ( is, recalling (., (3.9 Li α,0 ( [ 4 CAT (/4] = Li α,0 ( (CAT(/4 = Li α,0 ( CAT(/4. Now, using Theorem of [7], CAT(/4 = ( / + O( = + ζ(3/ Γ( / Γ( / Li 3/,0( + O(,

11 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES so that (3.9 is 0 α < 0, Γ( / Li 3 α,0( + O( α + O( ɛ α = 0, O( α > 0. When 3 α < this is O( α+ ; when 3 α, it is O(. In either case we get a contribution which is O( α ++(α ɛ. Hence [ ( C C R ( = + C Γ(α + ] 4 Γ(( α + ( α + + =, < + O( + O( α ++(α ɛ = C ( α + + O( α ++(α ɛ, with the C s defined by the recurrence (3.8. Now using (3.6, the claim follows. Next, we handle the case when /4 < α < /. Proposition 3.. In the notation introduced above Proposition 3., when /4 < α < /, M ( = C ( (α+ + + O( (α+ +, for, the C s are defined by the recurrence (3.8. Proof. For = the claim is true as shown in Section 3. with C defined at (3.8. We will now analye each term in (3.7 for in a manner similar to that in the proof of Proposition 3.. First we consider the case that and are both nonero. It follows from the induction hypothesis that 4 M ( M ( = C C ( (+α + + O( (+α If 3 = 0 then the corresponding contribution to R ( is ( C C ( α + + O( α If 3 0 we use Lemma.4 to express 4 M C C ( M ( = 4Γ(( + α Li ( + α +,0( + O( (+α + 3 C C [( + α < ] ζ( ( + α + 4 Γ(( + α. The contribution to R ( is then (,, 3 times: C C 4Γ(( + α Li α + 3 +,0( + Li 3α,0( O( (+α + 3. Now 3 so α <. Hence the contribution when 3 0 is O( α = O( α + 3

12 JAMES ALLEN FILL AND NEVIN KAPUR by another application of part a of Theorem.6. Next we consider the case when is nonero but = 0. In this case using the induction hypothesis we see that 4 M ( M ( = 4 CAT(/4 M ( = C ( + α + O( α +. Applying Lemma.4 to the last expression we get 4 M C ( M ( = Γ( α Li α + 3,0( The contribution to R ( is hence ( times: + O( α + C [ α < ]ζ( α + 3 Γ( α C Γ( α Li α ,0( + Li 3α,0( O( α +. Using the fact that α > /4 and 3, we conclude that α < so that the contribution is O( α = O( α + 3 by part a of Theorem.6, unless 3 =. When 3 = we get a contribution of ( C Γ(α Γ(( α α + + O( α + 3, ( since and α > /4 imply α > 3/. The case when is nonero but = 0 is handled similarly. The final contribution comes from the single term both and are ero. In this case the contribution to R ( is Li α,0 ( [ 4 CAT (/4] = Li α,0 ( (CAT(/4 = Li α,0 ( CAT(/4 which is easily seen to be O( α + 3. Hence, as in the proof of Proposition 3., we see that R ( = C ( (α+ + + O( (α+ +3 with the C s defined by the recurrence (3.8, and the claim follows using ( Large toll functions (α /. When α / there is no need to apply the centering techinques. Define µ n ( := EXn and µ n ( := β n µ n (/4 n. Let M ( denote the ordinary generating function of µ n ( in n. Observe that M 0 ( = CAT(/4. As earlier, conditioning on the ey stored at the root, we get, for, r n ( := 4 µ n ( = n + + 3=, < β n j 4 n j µ j ( + r n (, n, (,, 3 b 3 n n µ j ( µ n j (,

13 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 3 for n and r 0 ( := µ 0 ( = µ 0 ( = 0. Let R ( denote the ordinary generating function of r n ( in n. Then and (3.0 R ( = + + 3=, < M ( = R ( (,, 3 (B( 3 [ 4 M (M ( ]. We can now state the result about the asymptotics of the generating function M when α > /. The case α = / will be handled subsequently, in Proposition 3.4. Proposition 3.3. Let ɛ > 0 be arbitrary, and define α < α < (3. c := ɛ α = α >. Then the generating function M ( of µ n ( satisfies M ( = C ( (α+ + + O( (α+ + +c for, the C s are defined by the recurrence (3.8. Proof. The proof is very similar to those of Propositions 3. and 3.. We present a setch. The reader is invited to compare the cases enumerated below to those in the earlier proofs. When = the claim is true by (3.5. We analye the various terms in (3.0 for, employing the notational convenience α := α +. When both and are nonero then the contribution to R ( is 4 ( C C ( α + + O( α +c+ when 3 = 0 and is O( α +c+ otherwise. When is nonero and = 0 the contribution to R ( is C Γ(α Γ(( α α + + O( α +c+ ( when 3 = and O( α +c+ otherwise. The case when is nonero and = 0 is identical. The final contribution comes from the single term when both and are ero. In this case we get a contribution of O( α+ which is O( α +c+. Adding all these contributions yields the desired result. The result when α = / is as follows. Recall that L( := log((. Proposition 3.4. Let α = /. In the notation of Proposition 3.3, M ( = ( + C,l L l ( + O( + ɛ l=0

14 4 JAMES ALLEN FILL AND NEVIN KAPUR for and any ɛ > 0, the C,l s are constants. The constant multiplying the lead-order term is given by (3. C,0 = (! (!π /. Proof. For = the claim is true by (3.4. For we estimate each term in (3.0. For this paragraph, suppose that and are both nonero. Then using the induction hypothesis we get (3.3 4 M (M ( = ( (++ 4 l=0 m=0 C,lC,mL (+ (l+m ( + O( (++3 ɛ. If 3 = 0 the corresponding contribution to R ( is ( ( + A,l L l ( + O( +3 ɛ, 4 l=0 A,0 = C,0C,0 and (A,l l= can be determined easily from (3.3. If 3 0 then we use Lemma.4 once again to get 4 M + (M ( = Ã +,l Li (+ +, + l(+o( (++3 ɛ, l=0 (Ã +,l + l=0 can be determined using (3.3 and Lemma.4. The corresponding contribution to R ( is then (,, 3 times: + l=0 Ã +,l Li ( + Li + 3 +, + l 3 3 O( (++ ɛ.,0( 3 + Invoing Lemma.3 we see that the contribution when 3 0 is O( + ɛ, which is O( + 3 ɛ. Next suppose that is nonero but = 0. In this case using the induction hypothesis and we see that 4 M (M ( = ( + By Lemma.4 this is l=0 l=0 C,lL l ( + O( + ɛ. B,l Li + 3, l( + O( + ɛ, (B,l l=0 are constants. The corresponding contribution to R ( is then ( times: l=0 B,l Li , l( + Li 3,0( O( + ɛ.

15 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES When 3 the contribution is O( + ɛ, which is O( +3 ɛ. When 3 = the contribution to R ( is ( + l=0 B,l L l ( + O( +3 ɛ, for some suitably defined constants B,l. Note that there is no lead-order contribution from this term. The terms = 0 and is nonero are similarly handled. The final contribution from the single term with = = 0 is O( +, which is O( +3 for. Summing all the contributions we get, for, R ( = ( + C,l L l ( + O( + 3 ɛ, C,0 = 4 C,l = 4 = = ( ( l=0 C,0C,0, C,0 = π, A,l + B,l, l. The recurrence for C,0 can be easily solved to yield [ ( C,0 =! [ ] ] / (! = π (!π. / The result follows immediately Asymptotics of moments. For 0 < α < /, we have seen in Propositions 3. and 3. that the generating function M ( of ˆµ n ( = β n µ n (/4 n satisfies M ( = C ( (α+ + + O( (α+ + +c, c := min{α ɛ, /}. By singularity analysis [9], Recall that 3 β n µ n ( n (α+ 4 n = C Γ((α + + O(n(α+ β n = 4n ( πn 3/ + O( n, 3 c. so that C π (3.4 µ n ( = Γ((α + + O(n (α+ c. n(α+ For α > / a similar analysis using Proposition 3.3 yields C π (3.5 µ n ( = Γ((α + + O(n (α+ c, n(α+

16 6 JAMES ALLEN FILL AND NEVIN KAPUR with now c as defined at (3.. Finally, when α = / the asymptotics of the moments are given by ( (3.6 µ n ( = π (nlog n + O(n (log n The limiting distributions. In Section 3.4. we will use our moment estimates (3.4 and (3.5 with the method of moments to derive limiting distributions for our additive functions. The case α = / requires a somewhat delicate analysis, which we will present separately in Section α /. We first handle the case 0 < α < /. (We assume this restriction until just before Proposition 3.6. We have (3.7 µ n ( = E X n = E[X n C 0 (n + ] = C π Γ(α nα+ + O(n α+ c with c := min{α ɛ,/} and µ n ( = E X n = C π Γ(α + + O(n α+ c. nα+ So (3.8 Var X n = Var X n = µ n ( [ µ n (] = σ n α+ + O(n α+ c, (3.9 σ := C π Γ(α + C π Γ (α. We also have, for, [ ] Xn (3.0 E = µ n( C π = n α+ n (α+ Γ((α + + O(n c. The following lemma provides a sufficient bound on the moments facilitating the use of the method of moments. Lemma 3.5. Define α := α +. There exists a constant A < depending only on α such that for all. C! A α Proof. The proof is by induction. For the basis case =, we simply choose A large enough so that the claim is true. For the induction step, assume. Define s := C /!. Dividing (3.8 by! we get (3. s = 4 Γ(α s j s j + s Γ(α α. Using Stirling s approximation we can find a constant γ < depending only on α so that, for all, Γ(α Γ(α α γα.

17 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 7 Using this in (3. we get By the induction hypothesis s A 4 s s j s j + γ s α 4 [j j ( j j ] α + γa ( α ( α. Now x x ( x x decreases as x increases for 0 < x / so that we can bound the sum above by the sum of two times the j = term and times the j = term. This and trivial bounds yield s A [( + 4( ] α A [ + ] α + A γ A α = A [3 α = (3 α α + γ A + A γ A α ] A α A α, + γa α the last inequality justified for all by choosing A large enough. This proves the claim. It follows from Lemma 3.5 and Stirling s approximation that (3. C π!γ((α + B for large enough B depending only on α. Using standard arguments [, Theorem 30.] it follows that X n suitably normalied has a limiting distribution that is characteried by its moments. Before we state the result, we observe that the argument presented above can be adapted with minor modifications to treat the case α > /, with X n replaced by X n. We can now state a result for α /. We will use the notation L to denote convergence in law (or distribution. Proposition 3.6. Let X n denote the additive functional on Catalan trees induced by the toll sequence (n α n 0. Define the random variable Y n as follows: X n C 0 (n + 0 < α < /, Y n := n α+ X n α > /, Then C 0 := n α+ n=0 n α β n 4 n, β n = ( n. n + n Y n L Y ;

18 8 JAMES ALLEN FILL AND NEVIN KAPUR here Y is a random variable with the unique distribution whose moments are (3.3 EY = C π Γ((α +, the C s satisfy the recurrence C = ( Γ(α + C j C j + 4 j Γ(( α + C, ; C = Γ(α. π The case α = / is handled in Section 3.4., leading to Proposition 3.9, and a unified result for all cases is stated as Theorem 3.. Remar 3.7. We now consider some properties of the limiting random variable Y Y (α defined by its moments at (3.3 for α /. (a When α =, setting Ω := C / we see immediately that EY = Γ( / Γ((3 / Ω, ( Ω = Ω j Ω j + (3 4Ω, Ω = j. Thus Y has the ubiquitous Airy distribution and we have recovered the limiting distribution of path length in Catalan trees [, 3]. The Airy distribution arises in many contexts including paring allocations, hashing tables, trees, discrete random wals, mergesorting, etc. see, for example, the introduction of [8] which contains numerous references to the Airy distribution. (b When α =, setting η := Y/ and a 0,l := l C l, we see that π Eη l = (5l / Γ((5l / a 0,l, a 0,l = l ( l a 0,j a 0,l j + l(5l 4(5l 6, a 0, =. j We have thus recovered the recurrence for the moments of the distribution L(η, which arises in the study of the Wiener index of Catalan trees [3, proof of Theorem 3.3 in Section 5]. (c Consider the variance σ defined at (3.9. (i Figure 3., plotted using Mathematica, suggests that σ is positive for all α > 0. We will prove this fact in Theorem 3.. There is also numerical evidence that σ is unimodal with max α σ (α. = achieved at α. = (Here. = denotes approximate equality. (ii As α, using Stirling s approximation one can show that σ ( α. (iii As α 0, using a Laurent series expansion of Γ(α we see that σ 4( log α.

19 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES Figure 3.. σ of (3.9 as a function of α. (iv Though the random variable Y (α has been defined only for α /, the variance σ has a limit at α = /: (3.4 lim α / σ (α = 8log π π. (d Figure 3. shows the third central moment E[Y EY ] 3 as a function of α. The plot suggests that the third central moment is positive for each α > 0, which would also establish that Y (α is not normal for any α > 0. However we do not now a proof of this positive sewness. [Of course, the law of Y (α is not normal for any α > /, since its support is a subset of [0,.] Figure 3.. E[Y EY ] 3 of Proposition 3.6 as a function of α.

20 0 JAMES ALLEN FILL AND NEVIN KAPUR (e When α = 0, the additive functional with toll sequence (n α = n is n for all trees with n nodes. However, if one considers the random variable α / Y (α as α 0, using (3.3 and induction one can show that α / Y (α converges in distribution to the normal distribution with mean 0 and variance 4( log. (f Finally, if one considers the random variable α / Y (α as α, again using (3.3 and induction we find that α / Y (α converges in distribution to the unique distribution with th moment! for =,,... In Remar 3.8 next, we will show that the limiting distribution has a bounded, infinitely smooth density on (0,. Remar 3.8. Let Y be the unique distribution whose th moment is! for =,,... Taing Y to be an independent copy of Y and defining X := Y Y, we see immediately that X is Exponential with unit mean. It follows by taing logarithms that the distribution of log Y is a convolution square root of the distribution of log X. In particular, the characteristic function φ of log Y has square equal to Γ( + it at t (,. By exponential decay of Γ( + it as t ± and standard theory (see, e.g., [4, Chapter XV], log Y has an infinitely smooth density on (,, and the density and each of its derivatives are bounded. So Y has an infinitely smooth density on (0,. By change of variables, the density f Y of Y satisfies f Y (y = f log Y (log y. y Clearly f Y (y is bounded for y not near 0. (We shall drop further consideration of derivatives. To determine the behavior near 0, we need to now the behavior of f log Y (log y/y as y 0. Using the Fourier inversion formula, we may equivalently study e x f log Y ( x = e (+itx φ(tdt, π as x. By an application of the method of steepest descents [(7.. in [], with g 0 =, β = /, w the identity map, 0 = 0, and α = 0], we get f Y (y π log (/y as y 0. Hence f Y is bounded every. Using the Cauchy integral formula and simple estimates, it is easy to show that f Y (y = o(e My as y for any M <. Computations using the WKB method [] suggest (3.5 f Y (y (/π /4 y / exp( y / as y, in agreement with numerical calculations using Mathematica. [In fact, the rightside of (3.5 appears to be a highly accurate approximation to f Y (y for all y.] Figure 3.3 depicts the salient features of f Y. In particular, note the steep descent of f Y (y to 0 as y 0 and the quasi-gaussian tail.

21 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES Figure 3.3. f Y of Remar α = /. For α = /, from (3.6 we see immediately that [ ] ( ( Xn E = π + O. nlog n log n Thus the random variable X n /(nlog n converges in distribution to the degenerate random variable / π. To get a nondegenerate distribution, we carry out an analysis similar to the one that led to (3.4, getting more precise asymptotics for the mean of X n. The refinement of (3.4 that we need is the following, whose proof we omit: A( CAT(/4 = π ( / L( + D 0 ( / + O( ɛ, (3.6 D 0 = n= By singularity analysis this leads to n / [4 n β n π n 3/ ]. (3.7 EX n = π nlog n + D n + O(n ɛ, (3.8 D = π (log + γ + πd 0. Now analying the random variable X n π / nlog n in a manner similar to that of Section 3.. we obtain ( 8 (3.9 Var [X n π / nlog n] = π log π n + O(n 3 +ɛ. Using (3.7 and (3.9 we conclude that [ Xn π / ] nlog n D n E = o( n and [ Xn π / ] nlog n D n (3.30 Var n = 8 π log π = lim α / σ (α, σ σ (α is defined at (3.9 for α /. [Recall (3.4 of Remar 3.7.]

22 JAMES ALLEN FILL AND NEVIN KAPUR It is possible to carry out a program similar to that of Section 3. to derive asymptotics of higher order moments using singularity analysis. However we choose to sidestep this arduous, albeit mechanical, computation. Instead we will derive the asymptotics of higher moments using a somewhat more direct approach ain to the one employed in [5]. The approach involves approximation of sums by Riemann integrals. To that end, define (3.3 X n := X n π / (n + log(n + D (n +, and ˆµ n ( := β n 4 n+e X n. Note that X 0 = D, ˆµ n (0 = β n /4 n+, and ˆµ 0 ( = ( D /4. Then, in a now familiar manner, for n we find n β j ˆµ n ( = 4 j ˆµ n j ( + ˆr n (, now we define ˆr n ( := + + 3=, < ( n ˆµ j ( ˆµ n j (,, 3 [ π (j log j + (n + jlog(n + j (n + log(n + + πn / Passing to generating functions and then bac to sequences one gets, for n 0, n ˆµ n ( = (j + β j 4 j ˆr n j(. j=0 Using induction on, we can approximate ˆr n ( and ˆµ n ( above by integrals and obtain the following result. We omit the proof, leaving it as an exercise for the ambitious reader. Proposition 3.9. Let X n be the additive functional induced by the toll sequence (n / n on Catalan trees. Define X n as in (3.3, with D defined at (3.8 and D 0 at (3.6. Then m 0 =, m = 0, and, for, (3.3 m = 4 Γ( π Γ( ( J,, 3 := + + 3=, < x=0 E[ X n /n] = m + o( as n, ] 3 ( 3 m m π J,,,, πm 3, x 3 ( x 3 [xlog x + ( xlog( x] 3 dx. Furthermore X n /(n + L Y, Y is a random variable with the unique distribution whose moments are EY = m, 0.

23 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES A unified result. The approach outlined in the preceding section can also be used for the case α /. For completeness, we state the result for that case here (without proof. Proposition 3.0. Let X n be the additive functional induced by the toll sequence (n α n on Catalan trees. Let α := α +. Define X n as X n C 0 (n + Γ(α (n + α 0 < α < /, (3.33 Xn := Γ(α X n Γ(α (n + α α > /, Γ(α C 0 := n= n α β n 4 n. Then, for = 0,,,..., [ E Xn /n α ] = m + o( as n, m 0 =, m = 0, and, for, (3.34 m = 4 Γ(α π Γ(α ( ( Γ(α m m,, 3 Γ(α with + + 3=, < J,, 3 := 0 3 J + 4,,3 πm, x α 3 ( x α 3 [x α + ( x α ] 3 dx. Furthermore, Xn L Y, Y is a random variable with the unique distribution whose moments are EY = m. [The reader may wonder as to why we have chosen to state Proposition 3.0 using several instances of n +, rather than n, in (3.33. The reason is that use of n + is somewhat more natural in the calculations that establish the proposition.] In light of Propositions 3.6, 3.9, and 3.0, there are a variety of ways to state a unified result. We state one such version here. Theorem 3.. Let X n denote the additive functional induced by the toll sequence (n α n on Catalan trees. Then X n EX n Var Xn L W, the distribution of W is described as follows: (a For α /, W = ( Y C π, with σ := C π σ Γ(α Γ(α + C π Γ (α > 0,

24 4 JAMES ALLEN FILL AND NEVIN KAPUR Y is a random variable with the unique distribution whose moments are EY = and the C s satisfy the recurrence (3.8. (b For α = /, C π Γ((α +, W = Y σ, with σ := 8 π log π, Y is a random variable with the unique distribution whose moments m = EY are given by (3.3. Proof. Define W n := X n EX n Var Xn (a Consider first the case α < / and let α := α +. By (3.7, (3.35 EX n = C 0 (n + + C π Γ(α nα + o(n α. Since X n defined at (3.33 and X n differ by a deterministic amount, Var X n = Var X n. Now by Proposition 3.0, (3.36 Var X n = E X n (E X n = (m +o(n α (m +o(n α = (m +o(n α. So σ equals m defined at (3.34, namely, 4 Γ(α ( Γ(α π Γ(α Γ(α J 0,0,. Thus to show σ > 0 it is enough to show that J 0,0, > 0. But J 0,0, = x x 3/ ( x 3/ [x α + ( x α ] dx, which is clearly positive. Using (3.35 and (3.36, π W n = X n C 0 (n + C Γ(α nα + o(n α, ( + o(σn α so, by Proposition 3.6 and Slutsy s theorem [, Theorem 5.4], the claim follows. The case α > / follows similarly. (b When α = /, EX n = π nlog n + D n + o(n by (3.7 and ( 8 Var X n = π log π + o( n by (3.30. The claim then follows easily from Proposition 3.9 and Slutsy s theorem.

25 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 5 4. The shape functional We now turn our attention to the shape functional for Catalan trees. The shape functional is the cost induced by the toll function b n log n, n. For bacground and results on the shape functional, we refer the reader to [5] and [8]. In the sequel we will improve on the mean and variance estimates obtained in [5] and derive a central limit theorem for the shape functional for Catalan trees. The technique employed is singularity analysis followed by the method of moments. 4.. Mean. We use the notation and techniques of Section 3. again. Observe that now B( = Li 0, ( and (3. gives So CAT(/4 = Γ( / [Li 3/,0( ζ(3/] + ( ζ(/ Γ( / ( + O( 3/. B( CAT(/4 = Γ( / Li 3/,( + c + c( + O( 3 ɛ, c and c denote unspecified (possibly 0 constants. The constant term in the singular expansion of B( CAT(/4 is already nown to be C 0 = B( CAT(/4 = (log n β n = 4 n. n= Now using the singular expansion of Li 3/, (, we get B( CAT(/4 = C 0 ( / L( (( log( γ( / +O(, so that (4. A( CAT(/4 = C 0 ( / L( (( log γ+o( /. Using singularity analysis and the asymptotics of the Catalan numbers we get that the mean a n of the shape functional is given by (4. a n = C 0 (n + πn / + O(, which agrees with the estimate in Theorem 3. of [5] and improves the remainder estimate. 4.. Second moment and variance. We now derive the asymptotics of the approximately centered second moment and the variance of the shape functional. These estimates will serve as the basis for the induction to follow. We will use the notation of Section 3.., centering the cost function as before by C 0 (n +. It is clear from (4. that (4.3 M ( = L( (( log γ + O( /, and (3.7 with = gives us, recalling (., (4.4 R ( = C 0 + CAT(/4 Li 0, ( + 4Li 0, ( [ 4 CAT(/4 M (] + M (.

26 6 JAMES ALLEN FILL AND NEVIN KAPUR We analye each of the terms in this sum. For the last term, observe that / / as, so that M ( = L ( + 4(( log γl( + (( log γ + O( ɛ, the ɛ introduced to avoid logarithmic remainders. The first term is easily seen to be CAT(/4 Li 0, ( = K + O( ɛ, K := (log n β n 4 n. n= For the middle term, first observe that 4 CAT(/4 M ( = L( (( log γ + ( / L( + O( / and that L( = Li,0 (. Thus the third term on the right in (4.4 is 4 times: Li, ( + c + O( ɛ = L ( + γl( + c + O( ɛ. [The singular expansion for Li, ( was obtained using the results at the bottom of p. 379 in [7]. We state it here for the reader s convenience: Li, ( = L ( γl( + c + O(, c is again an unspecified constant.] Hence which leads to R ( = 8( log L( + c + O( ɛ, (4.5 M ( = 8( log ( / L( + c( / + O( ɛ. We draw the attention of the reader to the cancellation of the ostensible lead-order term L (. This ind of cancellation will appear again in the next section when we deal with higher moments. Now using singularity analysis and estimates for the Catalan numbers we get (4.6 µ n ( = 8( log nlog n + cn + O(n +ɛ. Using (4., Var X n = µ n ( µ n ( = 8( log nlog n + cn + O(n +ɛ, which agrees with Theorem 3. of [5] (after a correction pointed out in [8] and improves the remainder estimate. In our subsequent analysis we will not need to evaluate the unspecified constant c.

27 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES Higher moments. We now turn our attention to deriving the asymptotics of higher moments of the shape functional. The main result is as follows. Proposition 4.. Define X n := X n C 0 (n+, with X 0 := 0; µ n ( := E X n, with µ n (0 = for all n 0; and ˆµ n ( := β n µ n (/4 n. Let M ( denote the ordinary generating function of ˆµ n ( in the argument n. For, M ( satisfies with M ( = ( C l,0 = 4 / j=0 C,j L / j ( + O( + ɛ, l ( l C j,0 C l j,0, C,0 = 8( log. j Proof. The proof is by induction. For = the claim is true by (4.5. We note that the claim is not true for =. Instead, recalling (4.3, (4.7 M ( = L( (( log γ + O( /. For the induction step, let 3. We will first get the asymptotics of R ( defined at (3.7 with B( = Li 0, (. In order to do that we will obtain the asymptotics of each term in the defining sum. We remind the reader that we are only interested in the form of the asymptotic expansion of R ( and the coefficient of the lead order term when is even. This allows us to define away all other constants, their determination delayed to the time when the need arises. For this paragraph suppose that and. Then by the induction hypothesis (4.8 4 M ( M ( = ( 4 / + + / + l=0 A,,lL / + / l ( + O( ɛ, A,,0 = C,0C,0. (a If 3 = 0 then + = and the corresponding contribution to R ( is given by (4.9 4 ( ( + / + ( / l=0 A,,lL / + ( / l ( + O( +3 ɛ. Observe that if is even and is odd the highest power of L( in (4.9 is /. In all other cases the the highest power of L( in (4.9 is /. (b If 3 0 then we use Lemma.4 to express (4.8 as a linear combination of { Li + +,l ( } / + / l=0

28 8 JAMES ALLEN FILL AND NEVIN KAPUR with a remainder that is O( ɛ. When we tae the Hadamard product of such a term with Li 0,3 ( we will get a linear combination of { } / + / Li + ( +,l+ 3 and a smaller remainder. Such terms are all O( + + ɛ, so that the contribution is O( +3 ɛ. Next, consider the case when = and. Using the induction hypothesis and (4.7 we get (4.0 l=0 4 M ( M ( = / + ( j=0 B + j,jl ( + O( + ɛ, with B,0 = C,0. (a If 3 = 0 then = and the corresponding contribution to R ( is given by (4. ( ( / + + j=0 B,j L + j ( + O( +3 ɛ. (b If 3 0 then Lemma.4 can be used once again to express (4.0 in terms of generalied polylogarithms, whence an argument similar to that at the end of the preceding paragraph yields that the contributions to R( from such terms is ɛ, which is O( +3 ɛ. The case when and = is handled symmetrically. When = = then (/4 M ( M ( is O( ɛ and when one taes the Hadamard product of this term with Li 0,3 ( the contribution will be O( ɛ. Now consider the case when = 0 and. Now M 0 ( = CAT(/4, so that (4. O( 4 M ( M ( = ( / j=0 C,jL / j ( + O( + ɛ. By Lemma.4 this can be expressed as a linear combination of { } / Li ( +,j j=0 with a O( + ɛ remainder. When we tae the Hadamard product of such a term with Li 0,3 ( we will get a linear combination, call it S(, of { } / Li ( +,j+ 3 with a remainder of O( + ɛ, which is O( +3 ɛ unless =. When =, by Lemma.4 the constant multiplying the lead-order j=0

29 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 9 µ ( +, term Li +, ( in S( is C,0 0. When we tae the Hadamard product of this term with Li 0,3 ( we get a lead-order term of C,0 µ ( +, 0 Li +, +(. Now we use Lemma.3 and the observation that λ (α,r 0 µ (α,s 0 = to conclude that the contribution to R ( from the term with = 0 and = is (4.3 + ( + j=0 D,j L + j ( + O( +3 ɛ, with D,0 = C,0. Notice that the lead order from this contribution is precisely that from (4. but with opposite sign; thus the two contributions cancel each other to lead order. The case = 0 and is handled symmetrically. The last two cases are = 0, = (or vice-versa and = = 0. The contribution from these cases can be easily seen to be O( +3 ɛ. We can now deduce the asymptotic behavior of R (. The three contributions are (4.9, (4., and (4.3, with only (4.9 (in net contributing a term of the form ( + L / ( when is even. The coefficient of this term when is even is given by ( C,0C,0. 4 0< < even Finally we can sum up the rest of the contribution, define C,j appropriately and use (3.6 to claim the result A central limit theorem. Proposition 4. and singularity analysis allows us to get the asymptotics of the moments of the approximately centered shape functional. Using arguments identical to those in Section 3.3 it is clear that for µ n ( = C,0 π Γ( n/ [log n] / + O(n / [log n] /. This and the asymptotics of the mean derived in Section 4. give us, for, E [ ] X n nlog n C,0 π Γ(, E [ X n nlog n ] = o( as n. The recurrence for C,0 can be solved easily to yield, for, C,0 = (!(!!(! σ, σ := 8( log. Then using the identity Γ( ( [ (! = = π (! we get C,0 π Γ( = (!! σ. ]

30 30 JAMES ALLEN FILL AND NEVIN KAPUR It is clear now that both the approximately centered and the normalied shape functional are asymptotically normal. Theorem 4.. Let X n denote the shape functional, induced by the toll sequence (log n n, for Catalan trees. Then X n C 0 (n + nlog n L N(0,σ and C 0 := and σ := 8( log. n= X n EX n Var Xn (log n β n 4 n, β n = ( n, n + n L N(0,, 5. Sufficient conditions for asymptotic normality In this speculative final section we briefly examine the behavior of a general additive functional X n induced by a given small toll sequence (b n. We have seen evidence [Remar 3.7(d] that if (b n is the large toll sequence n α for any fixed α > 0, then the limiting behavior is non-normal. When b n = log n (or b n = n α and α 0, the (limiting random variable is normal. Where is the interface between normal and non-normal asymptotics? We have carried out arguments similar to those leading to Propositions 3.9 and 3.0 (see also [5] that suggest a sufficient condition for asymptotic normality, but our proof is somewhat heuristic, and further technical conditions on (b n may be required. Nevertheless, to inspire further wor, we present our preliminary indications. We assume that b n b(n, b( is a function of a nonnegative real argument. Suppose that x 3/ b(x is (ultimately nonincreasing and that xb (x is slowly varying at infinity. Then EX n = C 0 (n + ( + o( πn 3/ b (n, C 0 = n= b n β n 4 n. Furthermore, Var X n 8( log [nb (n] nlog n, and X n C 0 (n + nb (n L N(0,σ, σ = 8( log. nlog n This asymptotic normality can also be stated in the form X n EX n Var Xn L N(0,. References [] P. Billingsley. Probability and measure. John Wiley & Sons Inc., New Yor, third edition, 995. A Wiley-Interscience Publication. [] N. Bleistein and R. A. Handelsman. Asymptotic expansions of integrals. Dover Publications Inc., New Yor, second edition, 986. [3] H.-H. Chern and H.-K. Hwang. Phase changes in random m-ary search trees and generalied quicsort. Random Structures Algorithms, 9(3-4:36 358, 00. Analysis of algorithms (Krynica Morsa, 000.

31 LIMITING DISTRIBUTIONS FOR ADDITIVE FUNCTIONALS ON CATALAN TREES 3 [4] W. Feller. An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons Inc., New Yor, 97. [5] J. A. Fill. On the distribution of binary search trees under the random permutation model. Random Structures Algorithms, 8(: 5, 996. [6] J. A. Fill, P. Flajolet, and N. Kapur. Singularity analysis, Hadamard products, and tree recurrences, 003. Preprint. Available at [7] P. Flajolet. Singularity analysis and asymptotics of Bernoulli sums. Theoret. Comput. Sci., 5(-:37 38, 999. [8] P. Flajolet and G. Louchard. Analytic variations on the Airy distribution. Algorithmica, 3(3:36 377, 00. Mathematical analysis of algorithms. [9] P. Flajolet and A. Odlyo. Singularity analysis of generating functions. SIAM J. Discrete Math., 3(:6 40, 990. [0] P. Flajolet and J.-M. Steyaert. A complexity calculus for recursive tree algorithms. Math. Systems Theory, 9(4:30 33, 987. [] N. Fröman and P. O. Fröman. JWKB approximation. Contributions to the theory. North- Holland Publishing Co., Amsterdam, 965. [] H.-K. Hwang and R. Neininger. Phase change of limit laws in the quicsort recurrence under varying toll functions. SIAM J. Comput., 3(6:687 7, 00. [3] S. Janson. The Wiener index of simply generated random trees. To appear. Random Structures Algorithms. Available at [4] S. Janson. Ideals in a forest, one-way infinite binary trees and the contraction method. In Mathematics and computer science, II (Versailles, 00, Trends Math., pages Birhäuser, Basel, 00. [5] D. E. Knuth. The art of computer programming. Volume. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 3rd edition, 997. [6] Y. Koda and F. Rusey. A Gray code for the ideals of a forest poset. J. Algorithms, 5(:34 340, 993. [7] H. M. Mahmoud. Evolution of random search trees. John Wiley & Sons Inc., New Yor, 99. A Wiley-Interscience Publication. [8] A. Meir and J. W. Moon. On the log-product of the subtree-sies of random trees. Random Structures Algorithms, (:97, 998. [9] R. Neininger. On binary search tree recursions with monomials as toll functions. J. Comput. Appl. Math., 4(:85 96, 00. Probabilistic methods in combinatorics and combinatorial optimiation. [0] U. Rösler. A limit theorem for Quicsort. RAIRO Inform. Théor. Appl., 5(:85 00, 99. [] L. Taács. A Bernoulli excursion and its various applications. Adv. in Appl. Probab., 3(3: , 99. [] L. Taács. Conditional limit theorems for branching processes. J. Appl. Math. Stochastic Anal., 4(4:63 9, 99. [3] L. Taács. On a probability problem connected with railway traffic. J. Appl. Math. Stochastic Anal., 4(: 7, 99. address, James Allen Fill: jimfill@jhu.edu URL, James Allen Fill: address, Nevin Kapur: nevin@jhu.edu URL, Nevin Kapur: Department of Mathematical Sciences, The Johns Hopins University, 3400 North Charles Street, Baltimore MD 8, USA