On the Number of Unary-Binary Tree-Like Structures with Restrictions on the Unary Height

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1 Ann. Comb Published online February 3, The Authors This article is an open access publication Annals of Combinatorics On the Number of Unary-Binary Tree-Lie Structures with Restrictions on the Unary Height Olivier Bodini, Danièle Gardy 2, Bernhard Gittenberger 3, and Zbigniew Gołębiewsi 4 Institut Galilée, Université Paris, Nord, Villetaneuse, France olivier.bodini@lipn.univ-paris3.fr 2 DAVID Laboratory, University of Versailles Saint Quentin en Yvelines, 7840 Vélizy Villacoublay, France daniele.gardy@uvsq.fr 3 Institute for Discrete Mathematics and Geometry, Technische Universität Wien, Wiedner Hauptstrasse 8-0/04, A-040 Wien, Austria gittenberger@dmg.tuwien.ac.at 4 Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, Wybrzeże Wyspiańsiego 27, Wrocław, Poland zbigniew.golebiewsi@pwr.edu.pl Received October 5, 205 Mathematics Subject Classification: 05A6, 05A5, 60C05, 68N8 Abstract. We investigate various classes of Motzin trees as well as lambda-terms for which we derive asymptotic enumeration results. These classes are defined through various restrictions concerning the unary nodes or abstractions, respectively: we either bound their number or the allowed levels of nesting. The enumeration is done by means of a generating function approach and singularity analysis. The generating functions are composed of nested square roots and exhibit unexpected phenomena in some of the cases. Furthermore, we present some observations obtained from generating such terms randomly and explain why usually powerful tools for random generation, such as Boltzmann samplers, face serious difficulties ingenerating lambda-terms. Keywords: asymptotic enumeration, generating functions, lambda-terms, trees, directed acyclic graphs, nested square-roots, singularity analysis The first author s wor was supported by ANR Metaconc project France Part of the wor of the second author was done during a long-term visit at the Institute of Discrete Mathematics and Geometry of the TU Wien. She was partially supported by the P.H.C. Amadeus project Boolean expressions: compactification, satisfiability and distribution of functions and by the ANR project BOOLE Both the third author and the fourth author were supported by FWF grant SFB F The third author was also supported by ÖAD, grant F0/205.

2 46 O. Bodini et al.. Introduction This paper is mainly devoted to the asymptotic enumeration of lambda-terms belonging to a certain subclass of the class of all lambda-terms. Roughly speaing, a lambda-term is a formal expression built of variables and a quantifier λ which in general occurs more than once and acts on one of the free variables of the subsequent sub-term. Lambda calculus is a set of rules for manipulating lambda-terms and was invented by Church and Kleene in the 930s see [7, 39, 40] in order to investigate decision problems. It plays an important role in computability theory, for automatic theorem proving or as a basis for some programming languages, e.g., LISP. Due to its flexibility it can be used for a formal description of programming in general and is therefore an essential tool for analyzing programming languages cf. [4, 42] and is now widely used in artificial intelligence. Furthermore, in typed lambda calculus types can be interpreted as logical propositions and lambda-terms of a given type as proofs of the corresponding proposition. This is nown as the Curry-Howard isomorphism see [49] and constitutes in view of the above-mentioned lin to programming a direct relationship between computer programs and mathematical proofs. Recently, there has been rising interest in random structures related to logic in general see [20,29,30,53] and in the properties of random lambda-terms in particular see [9], [33, 34] or [43]. Although lambda-terms are related to Motzin trees, the counting sequences of these two objects have widely different behaviours. In this paper, a tree-lie behaviour is meant to be that the counting sequence asymptotically behaves as is typical for trees with average height asymptotically proportional to the square root of the tree size. See [22] for numerous results on such trees as well as many other classes of trees. For analyzing the structure of random lambda-terms it is important to now the number of lambda-terms of a given size. It turns out that this is a very hard problem. The reason is that there are many degrees of freedom for assigning variables to a given abstraction. This leads to a large number of lambda-terms of fixed size. If we translate the countingprobleminto generatingfunctions, then the resulting generating function has radius of convergence equal to zero. Thus, none of the classical methods of analytic combinatorics see [27] is applicable. Therefore, in this paper we study simpler structures, obtained by bounding either the total number of abstractions or by introducing bounds on the levels of nesting either globally or locally, to be formally defined in the next section of lambda-terms. Note that the number of nesting levels of abstraction or even the number of abstractions in lambda-terms which occur in computer programming is in general assumed low comparedto their size, for example, for implementing lambda-calculuswe need to bound the height of the underlying stac, which is determined by the maximal allowed number of recursions. Clearly, using more binary nodes applications in favour of unary nodes abstractions results in general in requiring fewer levels of recursion and therefore a smaller stac. Even more, Yang et al. [52], who developed the very successful software Csmith for finding bugs in real programs lie the gcc compiler, write on [52, p. 3]: Csmith begins by randomly creating a collection of struct type declarations. For each, it randomly decides on a number of members and

3 Number of Restricted Unary-Binary Tree-Lie Structures 47 the type of each member. The type of a member may be a possibly qualified integral type, a bit-field, or a previously generated struct type. A declaration in a C program corresponds to an abstraction in a lambda-term, and the engineers chose the number of abstractions before randomlygeneratingthe rest of the program. That means that they expect the number of abstractions to be independent of the size of the lambdaterm corresponding to their program. Thus, requiring bounds lie those mentioned above does not seem to be a severe restriction from a practical point of view. Preliminary results on the enumeration of lambda-terms with bounded unary height appeared in [6]. The plan of the paper is as follows: we present all the formal definitions of the objects of our interest in Section 2 and then, in Section 3, some results on restricted classes of Motzin trees forcomparisonpurposes. The enumerationof lambda-terms with a fixed or bounded number of unary nodes is done in Section 4. Sections 5 and 6 contain the main results of our paper. They are devoted to the enumeration of lambdaterms where all bindings have bounded unary length and lambda-terms with bounded unary height, respectively. In order to achieve our results, we first derive generating functionsforthe associated countingproblems, which areexpressed as a finite nesting of radicals. Thenwe study the radii ofconvergenceand the type oftheir singularities. This will eventually allows us to determine their number asymptotically, as their size tends to infinity. A comparison of the two classes of lambda-terms is discussed in Section 7. Finally, we investigate how our theoretical results fit with simulations and discover some challenging facts on the average behaviour of a random lambda-term in Section A Combinatorial Description for Lambda-Terms 2.. Representation as Directed Acyclic Graphs A lambda-term is a formal expression which is described by the context-free grammar T a T T λ a.t, where a is a variable. The operation T T is called application. Using the quantifier λ is called abstraction. Furthermore, each abstraction binds a variable and each variable can be bound by at most one abstraction. A variable which is not bound by an abstraction is called free. A lambda-term without free variables is called closed, otherwise open. In this paper we deal with the enumeration of α-equivalence classes of closed lambda-terms: two terms are α-equivalent if one term can be transformed into the other by a sequence of α-conversions. An α-conversion is the renaming of a bound variable in the whole term cf. [3]. Since the lambda-terms we consider are closed, this means the actual variable names are unimportant; only the structure of the bindings is relevant, e.g., we consider the terms λ x.x and λ y.y to be identical. Furthermore, note that neither application noriterated abstraction is commutative, i.e., in particular, the terms λ x.λ y.t and λ y.λ x.t are different if and only if at least one variable x or y appears in T.

4 48 O. Bodini et al. A lambda-term can be represented as an enriched tree, i.e., a graph built from a rooted tree by adding certain directed edges pointers. First we construct a Motzin tree, i.e., a plane rooted tree where each node has out-degree 0,, or 2, if the edges were directed away from the root. We denote by the terms leaves, unary nodes, and binary nodes, the nodes with out-degree 0,, and 2, respectively. In this tree each application corresponds to a binary node, each abstraction to a unary node, and each variable to a leaf. The fact that an abstraction λ binds a variable v is represented by adding a directed edge from the unary node corresponding to the particular abstraction λ to the leaf labelled by v. Therefore, each unary node x of the Motzin tree is carrying zero, one or more pointers to leaves taen from the subtree rooted at x; all leaves receiving a pointer from x correspond to the same variable, and each leaf can receive at most one pointer. The Motzin tree obtained from a lambda-term t by removing all pointers variable bindings is called the underlying tree of t. For instance, the terms λ x.x x λ y.y and λ y.λ x.x λ x.y correspond to the enriched trees T 0 and T in Fig., respectively. In particular, these terms are closed lambda-terms, because every variable is bound by an abstraction, i.e., every leaf receives exactly one pointer. T 0 T Figure : Two examples of lambda-terms or α-equivalence classes if the labels of the leaves are removed: each unary node corresponds to an abstraction λ x binding all leaves below it which are labelled by x. Binary nodes correspond to applications merging their two subtrees t and t 2 into the more complex structure t t 2. As mentioned in the introduction, our interest in the present paper is in lambdaterms with restrictions on the number of abstractions and on the number of nesting levels of abstraction, either locally or globally. The following definitions will allow us to state our restrictions more precisely. Definition 2.. Consider a lambda-term and its associated enriched tree T. The unary length of the binding of a leaf e by some abstraction v in T directed edge from v to e, denoted by l u e, is defined as the number of unary nodes on the path connecting v and e in the underlying Motzin tree. Definition 2.2. Consider a lambda-term and its associated enriched tree T. The unary height of a vertex v of T, denoted by h u v, is defined as the number of unary nodes on the path from the root to v in the underlying Motzin tree. The unary height of T, h u T, is defined by max h uv. We use the same notions for Motzin trees v vertex of T as well.

5 Number of Restricted Unary-Binary Tree-Lie Structures 49 In this paper, we will enumerate lambda-terms with a fixed number of unary nodes, with bounded unary length of the bindings, or with bounded unary height. Of course, other simplifications are possible, such as bounding the number of pointers for each unary node. Such terms are related to linear terms where each abstraction binds exactly one variable, also called BCI terms and affine terms where each abstraction binds at most one variable, also called BCK terms logics as introduced in [3, 36 38], and their enumeration was treated in [8] and generalizations can be found in [7] and [0]. For their relations to lambda-calculus, see, for instance, [35] Generating Functions Associated with Lambda-Terms For each class of lambda-terms we will enumerate the terms of a given size. The size of a lambda-term is the number of nodes in the corresponding enriched tree. It is defined recursively by x =, λ x.t = + T, S T = + S + T. Remar 2.3. It should be mentioned that other definitions of the size of a lambda-term can be foundin the literature. See [4,50] forthree differentwaysto definethe size, all of which are in some sense natural from the computer science point of view. These three concepts of size have been generalized in [3]. When regarding lambda-terms as enriched trees i.e., as particular directed acyclic graphs, the above definition is the most natural one, since it matches exactly the number of vertices of the graph. Further, we remar that the concept of size has significant impact on the counting sequence, its generating function, and the nature and location of the singularities of the generating function. The size concepts in [4, 3, 50] assign much larger sizes to lambda-terms than our concept, thus maing the counting sequence for general, unrestricted lambda-termsonly of exponentialgrowth and allowing a detailed asymptotic analysis. In order to count α-equivalence classes of lambda-terms of a given size we set up a formal equation which is then translated into a functional equation for generating functions using the well-nown symbolic method cf. [27]. Let us introduce the following atomic classes: the class of application nodes A, the class of abstraction nodes U, the class of free leaves F, and the class of bound leaves D. Then the class L of equivalence classes of lambda-terms can be described by the specification L = F + A L 2 + U subsf F + D, L, 2. where the term subsf F + D, L denotes the set of all lambda-terms which are built as follows: choose a lambda-term from L and some of its free leaves. The number of chosen leaves is arbitrary. Finally, replace all chosen free leaves by bound leaves. The last summand in 2. can be interpreted as the set of lambdaterms starting with an abstraction the unary node in U which binds exactly those leaves which have been replaced by the substitution operator subs.

6 50 O. Bodini et al. Remar 2.4. Note that the lambda-terms specified by L are not necessarily closed. Since α-conversion concerns only the bound variables, the equivalencehere is w.r.t. α-conversion and substitution of a free variable by another free variable which is not already present in the term. Remar 2.5. When investigating α-equivalence classes often the de Bruijn representation of lambda-terms is used. But this representation hides the enriched tree structure and since we count α-equivalence classes according to the size of their enriched tree, it is not suitable for our purpose. On the other hand, the substitution operator belongs to a collection of operations on combinatorial classes for which there exists a well-nown translation mechanism turning a symbolic equation lie 2. immediately into a functional equation for the corresponding generating functions for details, see [27, Ch. I III]. The specification 2. gives rise to a functional equation for the bivariate generating function Lz, f = z t f #free leaves in t, which reads as follows: t lambda-term Lz, f = f z + zlz, f 2 + zlz, f In particular, the formal generating function for lambda-terms without free variables is Lz, 0 = [ f 0] Lz, f = z 2 + 2z 3 + 4z 4 + 3z z z z z z 0 +. Note that these functional equations have to be considered in the framewor of formal power series since the fast growth of the coefficients of the generating function implies that the radius of convergence of Lz, 0 is zero see Corollary 6.20 below. Furthermore, note that the problem of counting closed or open lambda-terms is essentially the same. Indeed, the formal generating function for open lambda-terms can be derived from 2.2 the formula Lz, Lz, 0 = zlz,0 zlz,02 z. Consequently, the counting problems for lambda-terms with and lambda-terms without free variables are of the same difficulty and the solution for one of them yields the solution for the other. Before we start with the analysis of the generating functions associated with the considered combinatorial structures let us introduce a few further notions. Definition 2.6. We say that a function f : C C has a singularity of type α at z = ρ if there is a constant c such that f z c z α, ρ as z ρ inside the domain of analyticity of f.

7 Number of Restricted Unary-Binary Tree-Lie Structures 5 Definition 2.7. If f : C C is a function which is analytic at 0. Then let S denote the set of all singularities of f which lie on the circle of convergence of the Taylor series of f expanded at z = 0. Those singularities in S which are of smallest type are called the dominant singularities of f. Remar 2.8. It is well nown since Darboux [8] that the singularities on the circle of convergence determine the asymptotic behaviour of the coefficients of a series. The transfer theorems of Flajolet and Odlyzo [26] mae this much more precise and show that indeed only the dominant singularity in the sense of the definition above and its type yield the main term of the asymptotic behaviour. 3. Restricted Motzin Trees Beforeconsideringrestricted lambda-terms, we presentresults on classes of restricted Motzin trees. We shall considerclasses of Motzin trees with restrictions analogous to those for lambda-terms, namely, a fixed or bounded number of unary nodes, and a fixed or bounded unary height, where the unary height of a leaf is the number of unary nodes on the path from the root to that leaf, and the unary height of a tree is the maximal unary height of a leaf. The size of a Motzin tree is defined as the total number of nodes. The generating function associated with Motzin trees satisfies the functional equation Mz = z + zmz + zm 2 z. Solving this equation shows that the only power series solution is Mz = z 2z 3z 2z 2. The roots of the radicand are and /3, the latter being the dominant singularity of Mz and of type 2. Applying a transfer theorem from [26] yields that the number of Motzin trees of size n is asymptotically [z n ]Mz 3n+ 2 2n πn. 3.. Restrictions on the Total Number of Unary Nodes Let M q be the class of Motzin trees with exactly q unary nodes. We point out that a Motzin tree with exactly q unary nodes has a total size n equal to q + + 2m, where m is the number of binary nodes and m + the number of leaves. Proposition 3.. The number of Motzin trees of size n with exactly q unary nodes is 0 if n q mod 2; otherwise, it is asymptotically equivalent to n and for fixed q. 2 π 2 q q! 2n n q 3 2, as Proof. The assertion is an immediate consequence of Tutte s theorem [5] which implies directly that the number of Motzin trees of size n with exactly q unary nodes is n! q!n q /2!. For self-containedness and as it is in the flavour of this paper, we offer a proof of Proposition3. based on analytic combinatorics. Obviously, M 0 = C is the class of binary Catalan trees and its generating function is M 0 z =Cz = 4z 2 2z. For q, we have M q = U M q + q l=0 A M l

8 52 O. Bodini et al. M q l. This equation translates into a functional equation for the generating functions and we get after solving w.r.t. M q z M q z = zm q z + z l q M l zm q l z. 3. 2zM 0 z Lemma 3.2. There exists a sequence of polynomials P q z q 2 such that M q z = z q+ P q z 2, for q z 2 q 2 The polynomials P q z are given by the recurrence relation P 2 z = ; q 2 P q z = P q z + z P l zp q l z q l=2 Proof. For convenience, set = 2zM 0 z = 4z 2. From 3. one easily derives that M 2 = z 3 / 3 which fits with the assertion 3.2. We assume that the assumption M l = z l+ P l z 2 holds for l = 2,..., q ; Eq. 3. then gives M q z = z = z = z 2l = z q+ 2q + 2M zm q z + M q z z q P q z 2 q 2 + l=2 2q 2 + q 2 l=2 M l zm q l z q 2 l=2 2l M l zm q l z z l+ P l z 2 z q l+ P q l z 2 2q l P q z 2 + z 2 q 2 P l z 2 P q l z 2. From the last formula we read off the recurrence relation 3.3 and get the assertion after all. The asymptotic behaviour of the coefficients of M q z is now readily obtained recall that n = q + + 2m, with m being the number of binary nodes: [z n ]M q z = [ z 2m] P q z 2 = [z m P q z ] [z m ] 4m P q /4. 4z 2 q 2 4z q 2 z q 2 As [z m ] z q+ 2 3 mq 2 Γq 2, we get [z n ]M q z 2 n q P q /4 n q q q 2 3 Γq 2 P q/4 2 4 q Γq 2 2n n q 3 2. l=2

9 Number of Restricted Unary-Binary Tree-Lie Structures 53 Set a q = P q /4; then a 2 = a 3 = and a q = a q l q 2 a l a q l for q 4. This implies a q = 2 2 q C q where C q denotes the qth Catalan number. Plugging this into the asymptotic expression for [z n ]M q z gives immediately Proposition 3.. Next we consider the number of Motzin trees with at most q unary nodes, denoted by ˆM q. Then we have ˆM q z = 0 r q M r z and [z n ] ˆM q z = [z n ]M r z. 0 r q Hence, the last term of the sum gives the asymptotic main term which is [z n ]M q if n q mod 2, and [z n ]M q otherwise Restrictions on the Unary Height Define B as the set of Motzin trees such that all leaves are at the same unary height and ˆB as the set of Motzin trees where the leaves have unary height at most equal to All Leaves at the Same Unary Height Again, we start with setting up the specification and translating them into functional equations for the generating functions. Lemma 3.3. The class B 0 is equal to that of binary Catalan trees. Thus B 0 z = Cz = 4z 2 /2z. For, we have the recursive specification B = U B + A B B. Thus, the generating function associated with B satisfies B z = 4z 2z 2 B = 2z + 2z 2z + + 2z 4z 2z 2, where the second expression has + nested square roots. Now we turn to the asymptotic behaviour of such bounded unary height trees. For =, the dominant singularity of B z is at z = /2 and of type 4. The other singularity is at z = /2, but of type 2 and gives therefore an asymptotically negligible contribution. We obtain [z n ]B z n n 4 5 Γ. 3 4 Liewise, for 2, the singularities of B z are ±/2, which can easily be seen by induction. The singularity at z = /2 originates from the innermost radical only and is therefore of type /2. At z = /2 all radicals vanish at once and hence the singularity is of type /2 +. Consequently, as z /2, we have B z = 2 2 2z 2 + O 2z. Determining the asymptotic behaviour is now straightforward.

10 54 O. Bodini et al. Proposition 3.4. The number of Motzin trees in which all leaves have exactly unary height is [z n n+α n α ]B z 2 Γ α, with α = 2 +. Remar 3.5. This is another of the rather rare examples where the generating function of a recursively specified combinatorialstructure does not have a dominant singularity of type /2 or multiple of /2. A general discussion of possible singularity types of generating functions given by systems of functional equations was recently given by Banderier and Drmota [2] Motzin Trees of Bounded Unary Height The case = 0 again corresponds to binary Catalan trees and for larger a similar recursive specification as in the previous subsection holds. Lemma 3.6. The class ˆB is equal to that of binary Catalan trees. For, we have the recursive specification ˆB = Z + U ˆB + A ˆB ˆB. Thus, the generating function associated with B satisfies ˆB z = 4z 2z 2 4z 2 ˆB = 2z 4z 2z 2 + 2z 2z 4z z 4z, 2 where the second expression has + nested square roots. Again, the first function ˆB 0 z has the two singularities ±/2, but the next ones have different singularities. Indeed, the innermost square root has a zero at z = ±/2, but the next radical, 2z 4z 2 + 2z 4z 2, has a zero at ρ = < /2. The following few values are ρ 2 = , ρ 3 = , ρ 4 = Lemma 3.7. Let R 0 z = 4z 2 and R z = 2z 4z 2 + 2z R z. Then the values ρ, defined as the smallest real positive root of R z = 0, form a decreasing sequence. Proof. An easy inductive argument shows that the functions R z are decreasing functions on the positive real line of course, only up to their first singularity and smaller than one there: observe that for positive z we have R z + 4z. Notice that the class of Motzin trees of bounded unary height is a subclass of the class of unrestricted Motzin trees. The generating function of the latter one has dominant singularity equal to 3. Hence, for any fixed, we must have ρ 3. Now, suppose that ρ ρ. Then, since R z is decreasing for positive real z and R ρ = 0, we have R ρ = 2 ρ 4 ρ 2 0. But 2z 4z2 0 if and only if z ρ 3 for all. [ 5 4, 5 4 ] and 5 4 < /3 which contradicts the fact that

11 Number of Restricted Unary-Binary Tree-Lie Structures 55 Remar 3.8. Since the sequence ρ 0 is decreasing and bounded from below by 3, one can try to prove that ρ 3 as. Though numerical evidence supports this, it seems not obvious at all. Since it is not the ey point of our paper we decided to sip it. As ˆB z = /2z R z and each radical has a different dominant singularity, the dominant singularity of ˆB is at z = ρ and of type /2. Here the dominant singularity always comes from the outermost radical. Thus, we obtain the following result: Proposition 3.9. The number of Motzin trees with unary height at most equal to is [z n ] ˆB z C ρ n n 2 3, where ρ is defined in Lemma 3.7 and C is a suitable constant. 4. Enumeration of Lambda-Terms with Prescribed Number of Unary Nodes 4.. Recurrence for the Generating Functions We consider here the set S q of lambda-terms that have exactly q unary nodes. As a consequence their unary height is obviously bounded. We shall set up recurrence relations for the generating functions S q. Let z mar the total size and f mar the number of free leaves. The objects in S 0 are again binary Catalan trees and all the leaves are free since there is no unary node. Thus S 0 z, f = 4 f z 2. 2z For q =, either the unique unary node is equal to the root each leaf of the whole tree then either becomes bound or stays free or the root is a binary node and the unique unary node appears either in the left or in the right subtree. This yields the specification S = U subsf F + D, S 0 + A, S 0, S + A, S, S 0, and a recurrence relation for the generating function: S z, f = zs 0 z, f + + 2zS 0 z, f S z, f. 4. Solving, we get S z, f = zs 0z, f + 2zS 0 z, f = 4 f + z f z 2 For general q, a term has either a unary node as root and q unary nodes below or a binary node as root, and the q unary nodes are split into l nodes assigned to the left subtree, and q l nodes assigned to the right subtree. Hence, we obtain S q = U subs F F + D, S q + q l=0 A, Sl, S q l,

12 56 O. Bodini et al. which gives S q z, f = zs q z, f + + z q l=0 S l z, f S q l z, f. We can easily solve it and obtain S q z, f in terms of the S l z, f for l < q: q z S q z, f = S q z, f + + 2zS 0 z, f S l z, f S q l z, f. 4.2 The number of closed lambda-terms, which we are interested in, is then l= q S q z, 0 = S q z, + S l z, 0 S q l z, l= 4.2. Solving the Recurrence Lemma 4.. Let σ q f = 4 f + qz 2 for q 0. exists a rational function R q in q + variables such that Then, for all q 0, there S q z, f = z q σ q f 2 q l=0 σ l f + R q z, σ0 f,..., σ q f. 4.4 Moreover, the denominator of R q z, σ0 f,..., σ q f is of the form 0 l<q σ l f α l,q where the exponents α 0,q,..., α q,q are positive integers. Proof. The proof is based on induction on q. To start the induction, observe that S 0 z, f = σ 0z, f 2z and R 0 = 0. Now assume that 4.4 is true for S 0 z, f,..., S q z, f. Then by 4.2 and σ 0 f = 2zS 0 z, f, we have S q+ z, f = z z q σ q f + σ 0 f 2 q l=0 σ l f + + R q z, σ0 f +,..., σ q f + q + l= S l z, f S q l z, f By observing that σ q f + = σ q+ f, we obtain S q+ z, f = z zq σ q+ f σ 0 f 2 q l=0 σ l+ f + R q z, σ f,..., σ q f q + l= S l z, f S q l z, f. σ q is actually a function in the two variables z and f, but z plays no rôle in the statement and proof of this Lemma..

13 Number of Restricted Unary-Binary Tree-Lie Structures 57 The induction hypothesis implies that each S l z, f is itself a rational function of z, σ 0 f, σ f,..., σ l f. Hence, by setting R q+ = z q R q z, σ f,..., σ q f + σ 0 f S l z, f S q l z, f, l= we obtain S q+ z, f = z q σ q+ f 2 q l=0 σ l f + R q+ z, σ 0 f,..., σ q f. The expression of the denominator of R q comes readily from the recurrence expression. By setting f = 0, we obtain the following lemma: Lemma 4.2. The generating function enumerating all closed terms with exactly q unary nodes is S q z, 0 = z q 4qz 2 2 q + R q l=0 4lz 2 z,, 4z 2,..., 4q z 2, 4.5 where the rational function R q comes from Lemma 4.. Its dominant singularities are z = ± 2 q Asymptotics A lambda-term with exactly q unary nodes and i leaves has i binary nodes and size n = q+2i. From Lemma 4., the term R q will have singularities at z = ±/2 l for l < q. The first term on the right-hand side of 4.5 has singularities of smaller type at z = /2 q than the second term. Hence, it gives the dominant contribution to the asymptotics of [z n ]S q z, 0: [z n ]S q z, 0 [ z q+2i ] z q 4qz 2 [ z i] 4lz 2 2 q l= 4qz, 4lz 2 q l= as n tends to infinity. The denominator q l= 4lz contributes a multiplicative factor q l l= q = q q/2 q! and we obtain: Proposition 4.3. The number of closed lambda-terms with exactly q unary nodes and size n is 0 if n = q mod 2; otherwise its asymptotic value is 2 [z n ]S q z, 0 2 q q! πn 2 q n, as n. 3

14 58 O. Bodini et al. Remar 4.4. Though 3. and 4.2 have a very similar shape, the results of Propositions 3. and 4.3 are rather different. But note that even though 4.2 was the starting point, we eventually use 4.3 instead. Thus the resonance-lie behaviour induced by 3. and leading to the singularity of lower-order type described in Lemma 3.2 disappears Lambda-Terms with at Most q Unary Nodes We denote by T q z, f the generating function for lambda-terms with at most q unary nodes, where again z mars the nodes, and f the free leaves. If q = 0 we get once more the generating function for binary Catalan trees: T 0 z = S 0 z = Cz. Otherwise, T q z = q l=0 S lz and hence we can apply the results we obtained for a fixed number of unary nodes. The dominant singularity of T q z comes from S q z, whereas the terms S l z for l < q give negligible contributions to the asymptotics: the terms with exactly q unary nodes outnumber those with at most q such nodes and determine the asymptotic behaviour of the number of terms, which is the same for a fixed or bounded number of unary nodes. 5. Enumeration of Lambda-Terms with Bounded Unary Length of Bindings Now we turn our attention to the problem of enumerating lambda-terms with bounded unary length of their bindings for the definition, see Def. 2.. Let G denote the class of closed lambda-terms where all bindings have unary length less than or equal to. Our goal is to set up an equation specifying G. Define ˆP i, as the class of unary-binary trees such that every leaf e can be labelled in min{h u e + i, } ways. The classes ˆP i, can be recursively specified, starting from a class Z of atoms, by ˆP, = Z + A ˆP, ˆP, + U ˆP, and ˆP i, = iz + A ˆP i, ˆP i, + U ˆP i+,, for i <. Using again the traditional correspondence between specifications and generating functions we obtain ˆP, z = z z 2 4z 2 2z 5. and ˆP i, z = 4iz 2 4z 2 ˆP i+, z, 5.2 2z for i <. Note that for every positive integer, the class ˆP, consists of all Motzin trees with types of leaves. Moreover, the class ˆP 0, is isomorphic to the class

15 Number of Restricted Unary-Binary Tree-Lie Structures 59 G and thus the recursive specification gives directly the generating function G z = ˆP 0, z associated with G. We can rewrite 5.2 and 5. in the form ˆP i, z = 2z [i=] z ˆR i+, z, 5.3 where ˆR, z = z 2 4z 2, ˆR 2, z = 4 z 2 2z + 2z 2 + 2z ˆR, z, and ˆR i, z = 4 i + z 2 2z + 2z ˆR i, z, 5.4 for 3 i +. Hence, G z = ˆR +, z 2z. 5.. Analysis of the Radicands Let us now introduce the definition of a dominant radicand. Definition 5.. Consider a function f z which is analytic at z = 0, but not entire, and is of the form f z = p z + q z p z + q z... p z where p i z i =,..., and q i z i = 2,..., are polynomials in z. We call its j-th radicand, which is p z if j = and p j z + q j z... otherwise, a dominant radicand if it has a zero at a dominant singularity of f z. In order to proceed, we need to now the location and type of the dominant singularity ˆρ of the global generating function G z. This means actually that we need to now which radicands are dominant. Nested structures appear frequently in combinatorial objects. Often these structures lead to generating functions of the form of continued fractions see, for example, [6, 24]. Nested radicals are less frequent. They occur, for example, when enumerating binary non-planetrees [5,27,46], where there appears an iterated squareroot expansion. Lemma 5.2. For every > 0 and j +, the function ˆR j, z is strictly decreasing on the positive real line in the interval where it is defined as a real-valued function. 2z,

16 60 O. Bodini et al. Proof. We proceed by induction on j: ˆR, z = z 2 4z 2 is clearly decreasing for z real positive and > 0. Now assume ˆR j, z is decreasing for z > 0. Thus, for positive z we have d dz ˆR j, z = 8 j + z ˆR j, z + z d ˆR dz j, z. ˆR j, z The induction hypothesis implies that ˆR j, z ˆR j, 0 = and that d dz ˆR j, z < 0, which eventually gives d dz ˆR j, z < 8 j + z 0 for real positive z. Observe that the function ˆR,+ has the same dominant singularity as the function G z. Lemma 5.3. Assume > 0 and that the radical ˆR j, z has a positive singularity and let z 0 denote the smallest one. Then there are no complex singularities having the same modulus as z 0. Proof. From 5.3 we now that ˆR j, z = [ j=] z 2z ˆP j+, z 2. First, assume that z 0 is a root of ˆR j, z. Then 2z 0 ˆP j+, z 0 + j= z 0 =. If there were another complex root x = z 0 e iθ of the same modulus, then we would have = 2z 0 ˆP j+, z 0 + [ j=] z 0 = 2z 0 e iθ P j+, z 0 e iθ + j= z 0 e iθ. Since ˆP j+, z = n â j,,n z n can be viewed as the generating function of some suitable class of lambda terms, for all sufficiently large n we have â j,,n > 0. But this implies that 2z0 e iθ P j+, z 0 e iθ + j=z 0 e iθ <, whenever θ 0, which leads to a contradiction. If z 0 is not a root of ˆR j, z, then z 0 must be a zero of some ˆR j l, z with suitable l > 0. This follows from the nested structure 5.4 of the radicals. But then we can apply the arguments above to ˆR j l, z and arrive again at a contradiction. Let us now study the exact location and type of the dominant singularity of the functions G z. The next lemma will also prove that the singularity in the assumption of the previous lemma indeed exists. Lemma 5.4. Let ˆρ be the dominant singularity of the function G z. Then ˆρ = +2 comes from the innermost radicand and is of type 2. Proof. If a positive root of the radicand ˆR i, z exists, denote its smallest one as ˆρ i,. Let us consider the roots of the innermost radicand ˆR, z. Since ˆR, z is a quadratic equation, we now that it has two roots: +2 and 2. Moreover, since is a positive integer, ˆρ, = +2 is the dominant singularity of the generating function ˆP, z and of type 2.

17 Number of Restricted Unary-Binary Tree-Lie Structures 6 Let us now prove that none of the radicands ˆR j, z, 2 j +, has a positive root which is smaller than or equal to ˆρ,. By induction on j, using the formula ˆρ, = +2, and simplifying, we obtain ˆR 2, ˆρ, = 5 ˆρ, 2 > 0. Furthermore, from Lemma 5.2 we now that ˆR 2, z is decreasing on R +. Hence, ˆR 2, z does not have any positive root not larger than ˆρ,. Assume that ˆR j, z for some j 2 does not have any positive root smaller than or equal to ˆρ,. Then we get ˆR j+, ˆρ, = 4 j ˆρ 2, + 2 ˆρ, ˆR j, ˆρ, > 0 and again using the argument that ˆR j+, z is decreasing on the positive real line, we obtain that ˆρ, is the dominant singularity of ˆR j+, z and of type 2. Thus, ˆρ, is a dominant singularity of G z, and Lemma 5.3 implies that it is the only one. The following propositionwill be usefulto derive the asymptotic behaviourof the number of lambda-terms in the considered class of terms. Proposition 5.5. Let ˆρ be the root of the innermost radicand ˆR, z. Then, and ˆR, ˆρ ε = 2 ˆρ ε + O ε ˆR j, ˆρ ε = c j ˆρ ˆρ ε + Oε, j l=2 c l for 2 j +, where c = and c j = 4 j c j for 2 j +. Proof. Using the Taylor expansion of ˆR, z around ˆρ we obtain ˆR, z = ˆR, ˆρ + z ˆρ d dz ˆR, ˆρ + O z ˆρ 2. Knowing that ˆR, z has a zero at z = ˆρ and setting z = ˆρ ε we obtain the first claim 5.5. The next step is computing an expansion of ˆR j, z around ˆρ, where 2 j +. From 5.5 we conclude that ˆR, ˆρ ε = 2 ˆρ ε + Oε, and from the recursive relation 5.4 for ˆR j, z we have ˆR 2, ˆρ ε = 2 ˆρ + 6 ˆρ 2 4 ˆρ2 + 2 ˆρ 2 ˆρ ε + Oε. Using the formula ˆρ = +2 and simplifying, we get ˆR 2, ˆρ ε = 5 ˆρ ˆρ ε + Oε.

18 62 O. Bodini et al. Assume that for 2 j + we have ˆR j, = c j ˆρ 2 + d j ε + Oε. We just checed that this holds for j = 2 with c 2 = = 5 and d 2 = 4 ˆρ Now, we proceed by induction: observe that ˆR j+ ˆρ ε = 4 j ˆρ 2 ε2 2 ˆρ ε + 2 ˆρ ε c j ˆρ 2 + d j ε + Oε. Expanding, using the formula ˆρ = +2, and simplifying, we obtain ˆR j+ ˆρ ε = 4 j + 2 c j ˆρ 2 + d j c j ε + Oε. Setting c j+ = 4 j + 2 c j and d j+ = d j c j for 2 j, we obtain ˆR j+ ˆρ ε = c j+ ˆρ 2 + d j+ ε + Oε. Expanding d j+ using its recursive relation and d 2 = 4 ˆρ we have for 2 j d j+ = 4 ˆρ cl j l=2 We are now in the position to give the asymptotic behaviour of the number of lambda-terms having only bindings of bounded unary length. Theorem 5.6. Let, for any fixed, G z denote the generating function of lambdaterms where all bindings have unary lengths not larger than. Then, [z n ]G z 2 + 4π + j=2 c n n, as n, 5.6 j where c = and c j = 4 j c j, for 2 j Proof. Lemma 5.4 tells us that the dominant singularity ˆρ = +2 is algebraic and of type 2. Hence, we get the factor n 2 3 n + 2 in 5.6. Let us now consider the constant w.r.t. n term of 5.6. We have seen in Proposition 5.5 that for z close to ˆρ, and with the notions used in its proof, ˆR +, ˆρ ε = c + ˆρ 2 + d + ε + Oε. Since G z = 2z ˆR +, z, we get G ˆρ ε = ˆρ c+ d + 2 ˆρ 4 ˆρ 2 ε + Oε, c+

19 Number of Restricted Unary-Binary Tree-Lie Structures 63 which gives [z n ]G z d + 4 ˆρ 2 [z n ] zˆρ, as n. c+ Using the formulas d + = 4 ˆρ cl and ˆρ = l=2 the formula for the constant term. +2 and then simplifying, we obtain 5.2. Asymptotic Decrease of Constant Term Proposition 5.7. The multiplicative constant in 5.6 satisfies π + j=2 c = j D2 + e +! + O, as, where D = πωe 4 5 γ+ζ 4 2 and ω is a computable constant with numerical value ω The proofof Proposition 5.7 is focused on obtaining the asymptotic expansion of the product + j=2 c j for. Lemma 5.8. For M we have M j=2 where C is a suitable constant. c j = CM!4 M e 2M M O, M Proof. From the recursive relation 5.7 and by bootstrapping we obtain the asymptotic expansion c j = 4 j + 4 j 3 4 j j + O, as j, which we can rewrite as c j = 4 j +4 j 3+ω j, where ω j = Θ n 3 2. Consider now the product M j=2 c j for M large we shall tae M = + later on. We write it as M j=2 4 j + 4 j 3 M j=2 + ω j and consider each of the products separately. M j=2 + ω j: This product has a finite limit ω if the series j ω j is convergent, which is indeed the case. This limit can be computed numerically as c lim j M 2 j M 4 j+4. However, the convergence is slow. The best we have j 3 got from the numerical studies is ω = j 3 2

20 64 O. Bodini et al. M j=2 4 j + 4 j 3: This product gives us the asymptotic behaviour. Let us rewrite it as M j=2 M!4 M M + 3 = M!4 M exp j=2 j 4 j Now, nowing that log + j 4 3 j = j 4 5 j + O sum as 5 j 4 j + O = 2 M 5 4 H M + ζ 2 j 3 2 M log + 3. j=2 i 4 j j 3 2, we can compute our O M = 2 M 5 4 logm γ + ζ 2 + O, M where H M is the Mth harmonic number and γ = is the Euler-Mascheroni constant. We finally obtain M j=2 4 j + 4 j 3 = CM!4 M M 5 4 e 2 M where C = ω exp γ+ζ 2. + O M, Putting all pieces together we get the following formula for the constant term of π + j=2 c = j D2 + e + O, +! where D = πωe γ+ζ Enumeration of Lambda-Terms of Bounded Unary Height We now turn to the enumeration of lambda-terms with bounded unary height. Let H denote the class of closed lambda-terms with unary height less than or equal to. Our first goal is to set up an equation for the H. Define the class P i, as the class of unary-binary trees such that i + h u e for every leaf e i.e., the unary height of every leaf e is at most i and every leaf e is colored with one out of i + h u e colors. As in the previous section, we observe that P, is the class of all Motzin trees with types of leaves and P 0, is isomorphic to the class H. The class P, is isomorphic to the class obtained from H by allowing free leaves. This class in turn is isomorphic to the class of closed lambda-terms with a unary root: just add a unary

21 Number of Restricted Unary-Binary Tree-Lie Structures 65 node as new root to a term of the previous class and bind all free leaves by this newly added abstraction. For general i, P i, is isomorphic to the class of closed lambda-terms built as follows: consider a path of i unary nodes to which we append a Motzin tree with unary height less than or equal to i and call this structure the seleton. Then, for each leaf e there are i + h u e ways to bind it in order to mae a closed lambda-term out of the seleton. The classes P i, can be recursively specified, starting from a class Z of atoms, by P, = Z + A P, P, and, for i <, by and P i, = iz + A P i, P i, + U P i+,. 6. Translating into generating functions we obtain P, z = 4z 2 2z P i, z = 4iz 2 4z 2 P i+, z, 6.2 2z for i <. Due to the remars above, the recursive specification gives directly the generating function H z = P 0, z associated with H. We get an expression involving + nested radicals: H z = 2z + 2z 4 i + z 2 2z + 2z + 2z 4z 2 2z. 6.3 Note that for n we have [z n ]H z = [z n ]Lz, and thus H z converges to Lz, in the sense of formal convergence of power series cf. [27, p. 73]. In the next subsection we consider the singularities of this generating function and determine its dominant one together with its type we shall see that the location and the number of the dominant radicands changes with. Then we use this information to obtain the asymptotic behaviour of its coefficients. In Sections 3 and 5 we have seen examples where the dominant radicand is either the innermost one, the outermost one, or all radicands together. We now of no previous example where the position of the dominant radicand changes depending on the number of levels of nesting. 6.. Analysis of the Radicands

22 66 O. Bodini et al. We now consider how to determine the dominant singularity of the function H z: it is again built of nested radicals, hence its singularities are the values where at least one of the radicands vanishes. Theorem 6.4 below gives the dominant radicand in H z, i.e., the radicand whose zero is the dominant singularity of H z. But first, we introduce two auxiliary sequences which prove to be important in the sequel. Definition 6.. Let u i i 0 be the integer sequence defined by and N i i 0 by for all i 0. u 0 = 0, u i+ = u 2 i + i +, for i 0 N i = u 2 i u i + i, Corollary 6.2. The sequence N i i 0 can be written without reference to the sequence u i i 0 by N 0 = 0, N =, and N i+ = N 2 i +3N i +2+N i + 4N i 4i +, for i. Proof. Solve the equation N i = u 2 i u i + i, considered as a quadratic equation in u i, then plug its solution into the recursive definition for u i+. This requires a little care, as the choice of the solution for expressing u i in terms of N i differs for i = 0 and in the case i. Remar 6.3. Obviously, the two sequences u i i 0 and N i i 0 are strictly increasing and have super-exponential growth. Since the growth rate will be important for our analysis, we will turn to it later. Theorem 6.4. Let N i i 0 be the sequence defined in Def. 6. and let be an integer. Define j as the integer such that [N j, N j+. If N j, then the dominant radicand of H z is the j +th radicand counted from the innermost one outwards, and the dominant singularity ρ is of type 2. Otherwise, the jth and the j + st radicand vanish simultaneously at the dominant singularity of H z, which is equal to /2u j and of type 4. The rest of this section is devoted to the proof of Theorem The Radicands R i, Let us denote by R i, z the ith radicand i + of H z, according to the numbering from the innermost outwards as adopted in the assertion of Theorem 6.4, i.e., we have P i, z = R i+, z z We can write the radicands recursively as follows: R, z := 4z 2 and R i, z := 4 i + z 2 2z + 2z R i, z, 6.5

23 Number of Restricted Unary-Binary Tree-Lie Structures 67 for 2 i +, which gives R i, z = 4 i + z 2 2z + 2z 4 z 2 2z + 2z 4z 2. As H z = P 0, z = R +, z /2z, the dominant singularity of H z is the dominant singularity of R +, z as well The Dominant Singularity of a Radicand We show below that, for any fixed and for any i, i +, the ith radicand R i,, when restricted to the real part of its definition domain, is decreasing and use this to determine the interval where it is positive and to prove that it has a single real positive root, which turns out to be the dominant singularity. Lemma 6.5. For every > 0 and i +, the real function R i, z is strictly decreasing on the positive real line up to its first singularity. Proof. The proof is a simple inductive argument lie in Lemma 5.2. Corollary 6.6. For every > 0 and i +, the real function R i, z has at most one real positive root. Remar 6.7. If j and are such that [N j, N j+, then it will turn out that only the first j + radicands R, z,..., R j+, z will be relevant for our investigations. All of them have a real positive root. This holds due to the fact that R j+, z is a dominant radicand of H z, which we shall prove later on. Definition 6.8. Let j and be integers such that [N j, N j+. For i =,..., j+, let σ i, denote the smallest positive root of the radicand R i, z. Lemma 6.9. Assume that the radical R i, z has real positive singularities and let z 0 > 0 be the smallest of them. Then there are no complex singularities with modulus z 0. Proof. The proof is very similar to that of Lemma 5.3. Lemma 6.9 guarantees that H z can have only one dominant singularity, which must be on the positive real line. Now we turn our attention to the list σ i, i j where j is such that N j < N j+. Lemma 6.0. Let j and let [N j, N j+ be given and assume that σ i, and σ i+, exist. Then we have σ i+, σ i, for i j. Proof. First note that, if x 0 is a singular point of some radical, then it is also a singular point of all radicals which are lying more outwards. Therefore, if both functions R i, z and R i+, z have positive roots, then, by definition, σ i+, is the smallest positive root of R i+, z. Hence, it is a singularity of R i+, z and thus of R i, z as well. This immediately implies the assertion.

24 68 O. Bodini et al. Lemma 6.. For any i and, the inequality R i, z > R i,+ z holds for all z > 0 for which the two radicands are defined as real functions. Proof. Obviously, the assertion holds for i =. Then, observe R i, R i,+ = 4z 2 + 2z R i, R i,+ and hence an easy induction completes the proof When Two Successive Radicands Vanish Lemma 6.2. Assume that, for two indices j and such that j, the value σ j,, which is a root of R j,, is also a root of the radicand R j+,. Then σ j, = + +4 j. Moreover, R j p, σ j, = 4α p σ 2 j,, for all p < j, where the sequence α p is defined by { α0 = 0; α p = α p + p 2, f or p. Proof. By our assumption, the two successive radicands R j+, and R j, vanish for the same value z = σ j, = σ j+,. Therefore, from 6.5 shifted from j to j +, we obtain that 4 jz 2 2z = 0, and this can only happen if σ j, is equal to + +4 j. Now assume that j 2 and that z = σ j,, i.e., both R j, z and R j+, z are equal to 0. Then 0 = R j, z = 4 j + z 2 2z + 2z R j, z = 4z 2 + 2z R j, z, and thus R j, z = 4z 2. Going one step further and assuming that j 3, we obtain that R j, z = 4 j + 2z 2 2z + 2z R j 2, = 8z 2 + 2z R j 2,. Plugging the value R j, z = 4z 2 into this equation gives R j 2, z = 36z 2. We iterate and obtain for p j : R j p, z = 4 j + p + z 2 2z + 2z R j p, = 4p + z 2 + 2z R j p,. If R j p, z = 4α p z 2, then R j p, z = 4α p+ z 2 with α p+ = α p + p + 2.

25 Number of Restricted Unary-Binary Tree-Lie Structures 69 Remar 6.3. Note that Lemma 6.2 implies the existence of σ j,nj and σ j+,nj. By Lemma 6. we have σ j+, < σ j, and thus σ j+,nj +l exists for all l 0. This guarantees the existence of σ i, for all i j +, as we claimed in Remar 6.7. Lemma 6.4. If the values j and are such that there exists a value z cancelling both radicands R j+, and R j,, then we must have = N j where N j j 0 is defined by N 0 = 0 and N i := α i α i for i, with α i i 0 being the sequence defined in Lemma 6.2. Remar 6.5. The sequence N j j 0 in Lemma 6.4 is precisely the sequence defined in Def. 6.. Proof. From Lemma6.2, simultaneousvanishingofbothradicandsimplies that z = σ j,. Then we now the values of R j p, σ j, for all p = 0,..., j ; in particular, taing p = j gives R, σ j, = 4σ 2 j, α j. We have R, z = 4z 2, which implies that 4σ 2 j, = 4σ 2 j, α j. Hence, we have σ 2 j, = 4+α j. But we also now that a suitable value z = σ j, must be equal to + +4 j 4 j, which gives an equation for the integers and j involving also the sequence α j j 0 defined in Lemma 6.2: j = 4 j 4 + α j. 6.6 Setting l = j and solving gives l = j + α j j + α j, which leads to = j + α j 2 α j. Finally, the recurrence for α i see Lemma 6.2 gives = α j α j. The first values of the N j are given by Table. For each value = N j, the two radicands that vanish are those numbered by j and j +. Table : The first values N j and u j. j N j u j Lemma 6.6. No more than two radicands can vanish at the same positive value. If so, then these two radicands are consecutive ones. Proof. Assume that two non-consecutive radicands R i, and R j, vanish simultaneously. From Lemma 6.0, we now that the zeroes of the radicands decrease. Therefore, all the radicands R l, for i l j would vanish simultaneously. But it is not possible that more than two successive nested radicands R i,,..., R i+p, have a common positive zero: this can only happen for z = σ i,, but then the polynomial part 4 j + z 2 2z can be simplified into 4 j i z 2, hence it is strictly positive as soon as j > i +.

26 70 O. Bodini et al The Sequence N i i We establish here results about the growth of the sequence N i i. Lemma 6.7. The sequence u i i 0 defined in Def. 6. satisfies u i = α i +i. Moreover, the limit χ := lim u /2i i i exists. Furthermore, we have u i = χ 2i for sufficiently large i. As a consequence, both sequences u i i 0 and N i i have doubly exponential growth. Proof. The recurrence relation on the u i is clear from the definition of the α i in Lemma 6.2. Aho and Sloane [] study doubly exponential integer sequences x = x i i 0 of the form x i+ = x 2 i + g i with g i < x i /4 for i sufficiently large. They show there that for any such sequence x = x i the limit χ x := lim i x /2i i exists and that the sequence can be written, for i large enough, as x i = χx 2i. In our case it is easy to chec that, for i 4, g i = i + < u i /4. Hence, the sequence u i i 0 is of a form such that the result of [] applies, and lim i u /2i i can be numerically approximated by χ Finally, the relation N i = u 2 i u i + i = u i+ u i implies that N i i 0 is doubly exponential as well. Of course, since α i = u j+ j, the sequence α i i 0 is also doubly exponential. Remar 6.8. Note, however, that for neither of the sequences N i i and α i i 0 the result of Aho and Sloane [] can be applied see the recurrences they satisfy, given by Corollary 6.2 and Lemma The Singularities The following proposition sums up the properties of the singularities. Proposition 6.9. i Let ρ be the dominant singularity of H z for = 0,, 2,... Then the sequence ρ 0 is strictly decreasing. ii If there exists a j such that = N j, then the dominant singularity ρ Nj = σ j,nj = 2u j is a root of both radicands R j, and R j+,, and it is of type 4. iii For N j, N j+, the dominant singularity ρ is a root of the single radicand R j+, ; it is of type 2 and it lies in the interval 2u j+, 2u j. Proof. i If the jth radicand of H is dominant, then R j, ρ = 0. This implies that R j,+ ρ < R j, ρ = 0 and therefore ρ + < ρ, since the radicands are strictly decreasing functions by Lemma 6.5. ii If there exists a j such that = N j, then the pair j, = j, N j is a solution of 6.6. If we set σ j,nj = + +4N j j 4N j j, use 6.6, and then go bacwards the steps in the proof of Lemma 6.4, we eventually arrive at R j, σ j,nj = R j+, σ j,nj = 0.

27 Number of Restricted Unary-Binary Tree-Lie Structures 7 The type of the singularity is an immediate consequence of the fact that the two dominant radicands are consecutive ones. In order to obtain the last claim, note that N j j = α j α j j = α j + j 2 α j j and + 4N j j = 2α j + j 2 = 2u j 2, which gives, after simplification and choosing the root that is positive and has smallest modulus, σ j,nj = /2u j. iii For N j < < N j+, Lemmas 6.4 and 6.6 tell us that no two radicands vanish simultaneously; only the j + th radicand is the dominant one and the singularity is therefore of type /2. The bounds for σ j+, follow from the fact that for any given value of j the sequence of zeroes of R j, z is decreasing see Lemma 6. and Remar 6.3. The sequence of the dominant singularities for {N j j } is /2, /6, /24, /296, /4388, / , / ,... As a corollary, we get the well-nown result that Lz, 0 only converges at z = 0, which follows from [6] or the estimates given in [7, Section 5]. Corollary The radius of convergence of the generating function Lz, 0 enumerating all lambda-terms is zero. Proof. Obviously, the number of lambda-terms of size n is greater than or equal to the number the number of lambda-terms of size n and unary height p, for any p. Thus, for all the radius of convergence of Lz,0 must be smaller than or equal to the radius of convergence of H N z. But the sequence of these radii is the sequence 2u and converges to Asymptotic Analysis and Transition Between DifferentBehaviours Behaviour of the Radicands In order to proceed, we need some information on the behaviour of the radicands in a neighbourhood of the dominant singularity. This is done in the two propositions that follow: Proposition6.2 givesthe exact valuesand Proposition 6.22 their expansions at the singularity. Proposition 6.2. The values of R s,nj z at z = σ j,nj are as follows: i If s < j inner radicands, then, with the u j as defined in Lemma 6.7, u 2 j s R s,nj σ j,nj =. u j ii If s = j or s = j +, then R j,nj σ j,nj = R j+,nj σ j,nj = 0. iii If j + < s outer radicands, then λ s j R s,nj σ j,nj = u 2, j with the sequence λ l defined by λ 0 = 0 and λ l+ = l + + λ l for l 0.

28 72 O. Bodini et al. Proof. i The first assertion comes from Lemma 6.2, which gives R s, j σ j,nj = 4α j s σ 2 j,n j, and from Lemma 6.7, from which we have α j s = u 2 j s. ii The second assertion is simply the definition of σ j,nj. iii For the case s > j +, we first chec, using the equality 4N j jσ 2 j,n j 2σ j,nj = 0, that R j+2,nj σ j,nj = 4N j j σ 2 j,n j 2σ j,nj + 2σ j,nj R j+,nj σ j,nj = 4σ 2 j,n j. Now assume that for some l 2 we have R j+l,nj σ j,nj = 4λ l σ 2 j,n j and proceed by induction we have just checed that it holds for l = 2 with λ =. Then R j+l+,nj σ j,nj = 4N j j lσ 2 j,n j 2σ j,nj + 2σ j,nj R j+l,nj σ j,nj = 4σ 2 j,n j l + λ l = 4λ l σ 2 j,n j, again from the fact that 4N j jσ 2 j,n j 2σ j,nj = 0, and from the recurrence assumption on R j+l,nj σ j,nj. Proposition Let ρ = σ j,nj be the dominant singularity of H Nj z. Then, for any ε > 0 i R j,nj ρ ε = γ j ε + O ε 2, with γ j = d dz R j,n j ρ. 6.7 ii iii and for p 2, R j+,nj ρ ε = 2ρ γ j ε 2 + Oε, 6.8 R j+p,nj ρ ε = 4ρ 2 λ p + 2ρ 2 3 γ 4 j ε 4 + O ε 2, 2 p 2 p 2 i= λ i where the sequence λ i i is defined in Proposition 6.2. Proof. We now that R j,nj σ j,nj > 0 and that the function R j,nj z is analytic up to some value z > ρ. Hence, R j,nj z itself has a Taylor expansion around ρ which yields 6.7. Using the recurrence relation 5.4 for R j, z we immediately obtain 6.8.

29 Number of Restricted Unary-Binary Tree-Lie Structures 73 The next step is computing the expansion of R j+2,nj around ρ where it has a singularity of type 4. We obtain R j+2,nj ρ ε = 4ρ 2 + 2ρ 2ρ γ j ε 4 + O ε 3 4. Now consider the radicands R j+p,nj z for p 2 and proceed by induction: they have a common dominant singularity at z = ρ, which is of type 4. Thus, for all p 2, there exist a p 0 and b p such that R j+p,nj ρ ε = a p +b p ε 4 +O ε 2. We already now that a 2 = 4ρ 2 and b 2 = 2ρ 2ρ γ j. By the recurrence relation 5.4 for the radicands we get R j+p+,nj ρ ε = 4N j j pρ ε 2 2ρ ε+2ρ ε R j+p,nj ρ ε. Plugging in the expansion a p + b p ε 4 + O ε 2 for R j+p,nj ρ ε, expanding and simplifying the constant term through 4N j jρ 2 2ρ = 0 gives R j+p+,nj ρ ε = 4pρ 2 + 2ρ a p + ρb p ε 4 + O ε 2. ap Setting a p+ = 4pρ 2 +2ρ a p and b p+ = ρb p ap, we obtain R j+p+,nj ρ ε = a p+ + b p+ ε 4 + O ε 2. By dividing the recurrence for a p by 4ρ 2, we see that a p+ ap = p +. Coupled 4ρ 2 4ρ 2 with a 2 = 4ρ 2 and the definition of the λ i, this gives a p = 4ρ 2 λ p. Plugging the expression for a p that we have just obtained into the recurrence for the b p gives b p+ = b p and finally 2 λ p+ b p = b 2 2 p 2 p 2 i= λ i with b 2 = 2ρ 3 2 γ 4j Asymptotic Number of Lambda-Terms of Bounded Unary Height We are now in the position to give the asymptotic behaviour of the number of lambdaterms with bounded unary height. Theorem Let N i i 0 and u i i 0 be as in Definition 6.. i If there exists j 0 such that N j < < N j+, then there exists a constant h such that [z n ]H z h n 3/2 σ j, n, as n. 6.9

30 74 O. Bodini et al. ii If there exists j such that = N j, then the following asymptotic relation holds: where [z n ]H Nj z h n 5/4 σ j, n = hnj n 5/4 2u j n, as n, 6.0 γ /4 j 2u j /4 h Nj = 2 N j j+2, 6. 2Γ3/4 N j j i= λ i with γ j and the sequence λ i i 0 as defined in Proposition 6.2. Proof. The expressions given in 6.9 and 6.0 follow immediately from the fact that the dominant singularity for the cases N j and = N j is of type /2 and /4, respectively, and then applying the transfer theorem of Flajolet and Odlyzo [26]. What is left to do is proving 6.. ii If = N j, then apply Proposition 6.22 and the equation and get H Nj z = 2z H Nj ρ ε = a Nj j+ 2ρ R Nj +,N j z which gives using again the transfer theorem [26] b N j j+ 4ρ ε 4 + O ε 2, a Nj j+ [z n ]H Nj z b N j j+ 4ρ 2u j n [z n ]ρ z 4 a Nj j+ b N j j+ 4ρ 3 2u j n n a Nj j+ Γ 4. Finally, plug in the expressions of a Nj j+ and b Nj j+, then simplify using also Γ 4 = 4Γ 34, to obtain the expression of hnj The Location of Singularities for Large In this section we would lie to investigate the sequence ρ 0 itself. Let us first derive a few auxiliary results that we will need in order to proceed with the analysis of the asymptotic behaviour of ρ as. Proposition If ρ denotes the dominant singularity of H z, then ρ +2. Proof. Let us recall that G is the class of closed lambda-terms where all bindings have unary length less than or equal to, G z its generating function and ˆρ = +2 the dominant singularity of G z. Clearly, H G and therefore the exponential growth of G z is not larger than the exponential growth of H z, i.e., ρ ˆρ.

31 Number of Restricted Unary-Binary Tree-Lie Structures 75 Proposition For i = Ologlog, we have R i, + 2 = 2 i loglog + O, as. Proof. We prove the assertion by induction on i: R, = + 2 = O. Now, assume that for some i = Olog log the asymptotic relation R i, + 2 = 2 i loglog + O holds. Then R i+, i = i loglog + O. But it is easy to see that i = Ologlog: = + O, 4 i loglog = O Thus, we can finish the proof with the following calculations: loglog loglog R i+, + 2 = 2 i 2 + O 2 + O loglog = 2 i+ + O. Proposition If j is such that R j, z is a dominant radicand of the generating function H z, then j = Ologlog. Proof. Let us first consider the case where both the jth and the j + th radicand are dominant. From Theorem 6.4 we now that in that case = N j = u j+ u j. Moreover, from Lemma 6.7 we have u i = C 2i for sufficiently large i and with C Thus, = C 2 j+ + o and applying the logarithm twice on both sides of this equation we get j = Ologlog. In the case where R j, z is the only dominant radicand we have N j < < N j. It is enough to consider the left inequality N j = C 2 j +o <. Proceeding lie in the previous case we get j = Ologlog..

32 76 O. Bodini et al. We are now in the position to give the asymptotic behaviour of ρ. Theorem Let ρ be the dominant singularity of H z, then the asymptotic behaviour of ρ can be described as follows: If = N j R j, z and R j+, z are dominant, then ρ = 2 loglog 4 + O, as. 6.2 If N j < < N j only R j, z is dominant, then ρ = O, as j Proof. Let us first consider the case where = N j. From Lemma 6.2 and Proposition 6.9 we now that ρ = σ j, =. Proposition 6.26 tells us that + +4 j j = Ologlog and thus expanding yields ρ = O loglog, as desired. 3 2 Proving the result for the case where N j < < N j is less straightforward. Let us recall the result of Proposition 6.24: ρ ˆρ = +2 = O. So, 3 2 what is left is proving an upper bound. We have ρ < 2, which is the value that cancels the innermost radicand R,z = 4z 2. Unfortunately, this upper bound is too wea to be used in this proof. In order to improve the upper bound for ρ notice that ρ is a root of R j, z = 4 j + z 2 2z + 2z R j, z and that σ j, > ρ = σ j,. This inequality can be seen as follows: the wea inequality follows from Lemma 6.0. But it is even strict, because no two successive radicands can be zero. Thus the zeros σ j, and σ j, of the two respective radicands must be different. Furthermore, we now that R j, z is decreasing on the positive real axis see Lemma 6.5 and that ρ +2. Thus, for z [ρ, σ j, ] we have R j, +2 R j, z and R j, z R j, z where R j, z = 4 j + z 2 2z+ 2z R j, +2. One can easily chec that R j, z is decreasing for z > 0 and thus its positive root where κ = R j, +2 ρ = κ j + κ 2 κ,, must satisfy ρ ρ. This inequality together with ρ = O j where we used Proposition 6.25 for the asymptotic expansion of κ as, completes the proof.,

33 Number of Restricted Unary-Binary Tree-Lie Structures Exponential Decrease of the Constant Numerical computations for the coefficients of asymptotic expansions when =, 8, 35 give h = , h 8 = , h 35 = In Theorem 6.23 we presented an expression for these constants see Eq. 6. involving the quantities γ j and λ i i 0 which were defined in Proposition We now prove that the constant h Nj decreases exponentially fast as j. Proposition The constant h Nj satisfies, as j, e 2 u2 j u j h Nj = D 2u j u2 j u j + O, 6.4 u j where D = C / ω e 2 ζ/ /2 Γ3/4π /4 The proof of Proposition 6.28 starts from the value given in 6. and has two main parts: proving that γ j is of order u j and dealing with the product N j j i= λ i The Derivative of R j, z Maple computations show that γ j u j seems to converge quicly with a precision of 0 0 for j = 7 to a constant value, approximately equal to We will show that this indeed holds. Lemma Define w l,nj = d dz R l,n j ρ with ρ = σ j,nj as in the previous section. For p, set δ p, j = 4 N j p u j p+ and ε p, j =. u j u j 2u j p+ Then w,nj = 4 N j u j and, for p >, w p,nj = p δ s, j s= p ε r, j. r=s+ Proof. The computation of w,nj is straightforward from R,Nj z = 4N j z 2 and ρ = 2u j ; note that δ, j = 4 N j u j = w,nj. Now for p 2 we have R p,nj z = 4N j p + z 2 2z + 2z R p,nj z,

34 78 O. Bodini et al. which gives by derivation w.r.t z R p,n j z = 8N j p + z R p,nj z + z R p,n j z. R p,nj z Taing z = ρ = 2u j, we get w p,nj = R p,n j ρ = 4 N j p R p,nj ρ + u j w p,nj. 2u j R p,nj ρ Now we are computing γ j = w j,nj, i.e., we are interested in the w p,nj for p j. In this range, R p,nj ρ = u j p+ u j by Proposition 6.2, which gives w p,nj = 4 N j p + u j u j p+ + w p,n j = δ p, j + ε p, j w p,nj, u j 2u j p+ and it is then an easy exercise to obtain the explicit form of w p,nj. Then Set E s, p, j = p r=s+ w p,nj = ε r, j = p s= and we can now turn to γ j = w j,nj : we write γ j = = = j s= j s= δ s, j E s, j, j 2 p s j s l= j p+ u. l δ s, j E s, p, j 4 N j s u j s+ u j u j 4 N j + j + 2 u j and consider each term in turn. s= E s, j, j 4 j s= E s, j, j se s, j, j 2 u j j s= Lemma All the sums j s= E s, j, j, j s= se s, j, j, and j s= finite limit when j. u j s+ E s, j, j, u j s+ u j E s, j, j have a Proof. It suffices to write, e.g., the first sum as j s= and to remember the 2 j s j s l= u l exponential growth of the sequence u i i 0. The same argument holds for the second sum. Finally, since u j s+ < u j, the first sum is an upper bound of the last sum.

35 Number of Restricted Unary-Binary Tree-Lie Structures 79 This shows that γ j 4 N j u j E s, j, j, when j. The relation N j = u 2 j u j + j then gives readily the following lemma, where the value of the constant has been computed numerically. Lemma 6.3. The term γ j u j has a finite, nonzero limit when j : γ j s lim = C j u j Asymptotic Expansion of M i= λ i Lemma For M we have M i= λ i = 2π ω M M e M e M+ζ O, M for some computable constant ω which is numerically ω Proof. From the expression λ n = n + λ n and by bootstrapping, we obtain an asymptotic expansion for λ n when n + : λ n = n + n n 4n + O n, n which gives λ n = n + n ωn where ω n has order n 3 2. Consider now the product M n= λ n for M large we shall tae M = N j j later on. We can write it as M n= n+ n+ 2 M n= +ω n, and we consider separately each of the products. We first concentrate on the product of the terms + ω n. We now that it has a finite limit ω if the series n w n is convergent, which is indeed the case. This limit can therefore be computed as lim λ n. The convergence, however, M n M n+ n+ 2 is slow of order M. Thus the best we could achieve by numerical studies is ω We now turn to the product M n= n + n + 2, which gives the asymptotic behaviour. We begin by writing it as M! M n= + + = M! exp n 2n M log + +. n= n 2n Now M n= log + + n 2n = M n= n + O n, n

36 80 O. Bodini et al. where we can get effective bounds for the error terms. Observe that M n= O n n = O M. It remains to compute M n= n, which is equal to 2 M + ζ 2 + O M. We finally obtain M n= + + = e 2 M+ζ 2 + O n 2n M and the final result by Stirling s formula. Now inserting M = N j j = u 2 j u j into the formula of Lemma 6.32, we obtain N j j λ i = e ζ 2 2u 2 2 2π ω u j 2u j+ j e u2 j +2u j + O. 6.6 i= u j Putting All Together We now substitute Cu j for γ j in 6., according to Lemma 6.3, and also plug in the asymptotic equivalent for the product N j j i= λ i that comes from 6.6, to obtain 6.4 and 6.5 which finishes the proof of Proposition Bounded Unary Height vs. Bounded Unary Length of Bindings In Table 2 we give numerical results of the constant and exponential terms for the number of lambda-terms of bounded unary height and the number of terms where all bindings have bounded unary length. We can see that the exponential terms for growing are quite similar in both cases. Note that in case II the unary height is not bounded. Thus one might expect that bounding the unary height is a much stronger restriction and that therefore the exponential growth rates should exhibit a larger difference than they actually do. However, there is still a difference in the exponential growth rates, which maes it appear reasonable. The quotient of the exponential growth rates seems to tend to one which is as expected. The constant factors differ significantly in both cases, but they still share a common behaviour: they tend quite quicly to 0 as. One can also observe that for lambda-terms with bounded unary height in the cases where = N j not only the term n 5 4 appears instead of n 3 2, but also the constant factor behaves in a little different way: it is indeed smaller than one could expect. So far, we have no explanation for this behaviour. 8. Random Generation and Experiments As already mentioned in Section 2.2, the generating functions of closed and open lambda-terms general ones as well those in the restricted classes treated in this paper are lined by a very simple relation. Hence, Bolzmann samplers face the same difficulty when generating closed or open terms. Therefore, the following sections

37 Number of Restricted Unary-Binary Tree-Lie Structures 8 Table 2: Comparison of the values of the constant and exponential terms between terms of bounded unary height and terms with bounded unary length of bindings. Case I: Case II: Bounded unary height Bounded unary length of bindings constant term exp. term constant term exp. term will be devoted to the generation of closed terms only, since this seems to be the more important class. 8.. Random Generation of Closed Lambda-Terms To get a feeling of the average behaviour of a combinatorial object, a method of choice is the random generation of terms of large size. We considered two methods to try to generate a random lambda-term of bounded unary height: the recursive method [28] and Boltzmann sampling. Boltzmann samplers are powerful tools to generate objects in specified combinatorial classes uniformly at random. They were introduced in [23] and extended furthermore by numerous authors see, e.g., [3,4,25,48]. Note that, theoretically, a Boltzmann sampler can generatea tree of size close to n on average in linear time. We considered Boltzmann sampling of a closed term, with different success depending on the unary height: the efficiency decreases very quicly as the maximal unary height grows. When = 8, we can generate terms of size 0000 in a few seconds on a standard personal computer. Figure 3 presents a term of size 6853 with unary height bounded by 8. For large sizes and for the sae of readability, we have not indicated the edges between a unary node and the leaf labels.

38 82 O. Bodini et al. Figure 2: A random lambda-term of size 30, with the edges from unary nodes to leaves. However, if we consider lambda-terms with a maximal unary height of 35, a Boltzmann sampler is not able to produce objects of size larger than 200 in a reasonable time less than one day. The explanation of the phenomenon is as follows: an average random lambda-term begins with a large number of unary nodes; cf. Figure 5 see also [9] for a result in the same vein for a related model. Drawing the sufficient number of unary nodes has very low probability in the Boltzmann process. Figure 4 gives the various probabilities of drawing a leaf, a unary node, or a binary node, plotted against the unary height actually the number of recursive calls to the generator, but the design of the generator is such that a call is done if the unary height changes. After a long! starting phase, where the probability of stopping is larger than 0.9, the Boltzmann sampler becomes efficient. In other words, Boltzmann sampling is linear, but with a constant depending on the maximum unary height which grows very quicly: the recursive form of the specification of lambda-terms and their varying behaviour maes them not well amenable to random generation with a Boltzmann sampler. We have thus turned to the recursive method. Using the Maple pacage Combstruct, we have been able to generate quicly enough lambda-terms of size 200 and unary height bounded by 200 which means that there is de facto no restriction on the unary height of the lambda-term. Figure 5 shows what can be considered as a generic lambda-term for this size. Both classes, the one with bounded unary height and the one where all bindings have bounded unary length, can be used to approximate generic lambda-terms. But unfortunately, also in the case of bounded unary length of bindings we are facing the same difficulties when trying to generate them with a Boltzmann sampler. The probabilities for generating leaves, unary and binary nodes loo very similar to Figure 4. This fact can be explained as follows: for both classes of restricted lambda terms, the dominant radicand is either close or equal to the innermost radicand. But the Boltzmann sampler generates these from outside inwards. That is meant in the following sense: each square-root is the analytical analogue of the lifting from one unary level to the next cf. 6. and 6.2 in order to see this. The Boltzmann sam-

39 Number of Restricted Unary-Binary Tree-Lie Structures 83 Figure 3: The underlying Motzin tree of a random lambda-term of size 6853 and unary height 8 and its profile.

40 84 O. Bodini et al. Figure 4: Left: the probability that the singular Boltzmann sampler ΓP,35 of objects in P,35 stops immediately. Middle: the probability that the sampler ΓP,35 calls ΓP,35. Right: the probability that the sampler ΓP,35 independently calls 2 generators ΓP,35. Figure 5: Left: a lambda-term of size 200. Middle: its profile. Right: the average profile red computed over 500 random lambda-terms, compared with the average profile for plane binary trees blue: the Airy function. pler builds an object by starting from the root and attaching more and more nodes. So, the head of the term, i.e., the subtree comprising all nodes of unary height zero, is precisely the object corresponding to the outermost root; and this is generated before the nodes with larger unary height. But note that the generating function of the class of heads has a larger dominant singularity. Hence, the tuning parameter of the Boltzmann sampler is far away from this singularity, thus giving the sampler a strong bias towards stopping. On the other hand, moving the parameter into an interval where the sampler wors efficiently means that it is outside the domain of analyticity of the generating function associated with lambda-terms. This implies that we have a positive probability that the sampling process never stops. So the sampler becomeseven more inefficient than with the badly chosen tuning we had before moving it to the allegedly better region. Bodini et al. [2] developed a general framewor for Boltzmannsampling forwhich tuningparametersoutsidethe regionof convergenceofthe associated generating function can be used. This relies on anticipated rejection and might help to improve the Boltzmann samplers for generating random lambda-terms. For restricted Motzin trees the situation is totally different, because the dominant singularity comes from the outermost radicand. Thus the Boltzmann sampler starts to generate the object by generating subobjects corresponding to the root which

41 Number of Restricted Unary-Binary Tree-Lie Structures 85 determines the singularity, and we can choose the tuning parameter so that it lies in the optimal region, i.e., it lies close to the singularity of the associated generating function Shape of a Typical Lambda-Term Being able to draw repeatedly random lambda-terms allows us to mae tentative conjectures on their various parameters: profile, height, etc. Figure 6: Distribution of lambda-terms of size n and unary height, where n [,..., 98] and unary height [,..., 98]. We have plotted in Figure 6 the ratio between the number of lambda-terms with unary height exactly and size n, and the number of lambda-terms of size n without restriction on the height. The figure suggests that, for any given size n, the unary height is close to a Gaussian distribution. In particular, this gives some experimental explanation to the change of difficulty which we encountered when generating terms of small unary height size about 0000, unary height bounded by 8 and terms of fairly large unary height size about 0000, unary height bounded by 35: the wave indicates the good estimate for the number of abstractions in a lambda-term; for instance, if we consider lambda-terms of size 98, then the vast majority of these terms has a unary height between 25 and 50. Figure 5 shows a generic lambda-term, its profile number of nodes at each level and the profile averaged on 500 random lambda-terms, together with the average profile of a plane binary tree, which is up to scaling identical with that of Motzin trees since both tree classes are simply generated. From our simulations we can mae several empirical observations: The distribution of the profiles is poorly concentrated this is also the case for plane binary trees.