Efficient Implementation of Binary Trees in LISP Systems

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1 Efficient Implementation of Binary Trees in LISP Systems P. SIPALA Dipartimento di Eleltrotecnica, Eleltronica, Informatica, Universitd di Trieste, Italy In this paper, I consider how to implement the basic data strctre of LISP programming, the binary tree. I show that, for the compact encoding of LISP trees, three-pointer cells are preferable to the conventional two-pointer ones. The reslt is one I reported previosly, bt here the analysis is done more elegantly, sing the methods of probabilistic grammars. I also present a detailed implementation of LISP operators for the case in which mlti-pointer cells are adopted. Received December 986, revised November 987. INTRODUCTION The term 'software science' has been introdced by M. Halstead 7 to denote the statistical stdy of programming practice. This field incldes the empirical analysis of programming habits in the se of langage constrcts 8 and data strctres. 5 In general these stdies have shown that, in the composition of compter programs, there are strong reglarities, both in the lexical and in the strctral aspects of programs, which are broadly analogos to the ' laws' observed by Zipf in the analysis of natral-langage texts. 6 Sch reglarities (for example in the relative freqencies of identifiers and in the nesting of blocks) are of psychological interest, 2 and may also be sefl in software management and software engineering. For the stdy of tree strctres in programming, interesting data may be derived from the statistical measrements performed by Clark 54 on programs written in LISP, a programming langage in which binary trees are the fndamental data strctre. These measrements show that LISP trees occr statistically, in practical se, with a pecliar distribtion, where nbalanced strctres are prevalent. In a previos paper, 3 this observed distribtion of LISP trees was modelled as a branching process; 2 on this basis, it was shown that three-pointer cells are better than the conventional twopointer ones for the compact encoding of LISP data strctres. In this paper, I resme and extend the previos analysis, sing a different statistical model (probabilistic grammars) 5 in order to obtain a simpler treatment. I extend the discssion to cover the case of infinite trees, that is, list strctres containing loops. Moreover, I also examine in detail the implementation of the basic LISP operators for the case in which mltipointer cells are adopted. 2. MULTI-POINTER CELLS FOR LISP TREES Extensive measrements on LISP programs 54 have shown that there exists a marked bias in the freqency distribtion of binary trees, which are the fndamental data strctre in the langage (circlar lists and shared sblists are also allowed in most implementations, bt they occr rarely in practice). 3 Ordinarily, the left and right branches of tree nodes {car and cdr in LISP terminology) are sed for qite different prposes: cdr fields point to lists (that is, to sbtrees) three times as often as they do to terminal nodes (called atoms in LISP); car fields, on the other hand, point to lists less than one-third of the time. Becase LISP nodes are mainly chained together in the cdr direction, it might be convenient, for the compact encoding of LISP binary trees, to replace the conventional binary cells (with two pointers, car and cdr) by means of largerfc-arycells, each containing the k pointers denoted in LISP as car, ca(dfr, for i =,2,...,k-2, and c(rf)*"v. In this arrangement, which may be considered as a generalisation of cdr-coding, a A;-ary cell is a packed representation for a series of k binary cells, chained together in the cdr direction, with all inner links omitted and simply implied by the adjacency of fields, as shown in Fig.. This packed representation (generalised cdrcoding) has been shown by simlation 9 to provide considerable advantages for compaction. cadr caddr car cadr caddr Cell Cell 2 Cell 3 k-3 ca(d) r k-3 ca(d) r Cell k-2 k-2 k-\ ca(d)r c(d) r '*-2 ca(d) r c(d) r Cell Ar- k-ziy cell Figre I. k l binary cells, chained together in the cdr direction, and their representation as a single ft-ary cell. Downloaded from by gest on 3 September THE COMPUTER JOURNAL, VOL. 3, NO. 4, 988

2 EFFICIENT IMPLEMENTATION OF BINARY TREES IN LISP SYSTEMS There is obviosly a compromise to be fond on the size k of the -ary cells to be adopted. Large cells wold save many inner links, bt on the other hand they might often be mostly noccpied. Small cells wold be flly occpied more freqently, bt they wold also se mch more link space. My earlier reslt, 3 which will be proven here in a simpler way, asserts that k 3 is the best choice for typical LISP programs, providing an optimal storage cost abot 2 % lower than that of binary cells. The sections of this paper are organised as follows. I introdce, in Section 3, a model for the statistical distribtion of LISP binary trees. Trees are represented, in a standard way, by means of S-expressions (dotted pairs), which are generated sing a probabilistic context-free grammar. 5 The grammar is characterised by two parameters, p and q, indicating the probability of having an S-expression, instead of an atom, on the left (respectively, on the right) side of a dotted pair. Section 4 considers the case when p + q <, corresponding to the case of a consistent probabilistic grammar. 5 With this assmption, I derive an expression for c(k,p, q)- the expected nmber of pointers reqired to represent a tree - when k-ary cells are adopted. In Section 5 the discssion is extended to the case when p + q >. In this sitation the probabilistic grammar may generate not only finite trees, bt also trees having an infinite nmber of nodes. I deal first with the finite tree poplation, deriving for it an expression for c(k,p,q) similar to the one obtained for p + q < I. Section 6 examines the infinite trees which are seen as nfolded representations of linked strctres containing circlar paths. Section 7 discsses a possible implementation of the basic LISP primitives (car, cdr and cons) in terms of operations on linked fc-ary cells. 3. A PROBABILISTIC GRAMMAR The basic LISP data strctres, binary trees, are sally represented by means of symbolic expressions, called 5- expressions with dot notation. If a binary tree T consists of a terminal node, containing the atom, then the S- expression representing T, S(T), is simply the symbol. If Thas a left sbtree T x and a right sbtree T 2, then S(T) is obtained (recrsively) from 5(7]) and S(7^) by concatenating the two expressions, separated by a dot and enclosed in parentheses. For example, the binary tree shown in Fig. 2 is represented by means of the S- expression ((M.v).((w.(.(.v))).(.(.v)))). The seqel will be concerned mainly with the shape of binary trees and not with the contents (atoms) of their terminal nodes. However, left and right branches issing from nodes behave differently. The S-expressions will therefore se two distinct atomic symbols, and v, which will appear only on the left and only on the right side of dotted pairs, respectively. The S-expressions can be described by the context-free grammar G = (V N, V T,S,P), where V N = {S}, V T = {(,),.,, v} and the set P = {P x, P 2, P 3, P t } of prodctions is P = S-*(«.»); P 2 :S-*(.S); P 3 :S-+(S.v); Denote by p the probability that, in a dotted pair, the left-hand part is an S-expression, while p is the probability that it is an atom (the atom w); let q and q denote the corresponding probabilities for an S-expression and an atom (the atom v) on the right-hand side of dotted pairs. Measrements 5-4 indicate that, for typical LISP programs, p lies in the range , while q has vales in the range Assming that occrrences of atoms or S-expressions on therightand on the left of dotted pairs are independent, the grammar G above becomes a probabilistic grammar, * with probabilities attached to each prodction as follows: p 3 =p(l-q) It is now convenient to introdce the notation and theory for probabilistic grammars. 5 In particlar, denoting by \X\ the nmber of elements in a set X, the matrices Q (with \V N \ rows and P colmns), C (\P\ rows and \V N \ colmns), and T(\P\ rows and \V T \ colmns) are defined as follows. p }, if the yth prodction contains therth nonterminal on its left-hand side; -!, otherwise. C i} = nmber of occrrences of the yth non-terminal in the right-hand side of the /th prodction. T tf = nmber of occrrences of the /th terminal in the right-hand side of therth prodction. In the present case, since V N = {S}, V T = {(,),.,, v} and P is formed by the for prodctions in (), the matrices Q, C and T are () Downloaded from by gest on 3 September 28 Figre 2. The binary tree corresponding to the ^-expression ((II.!>).((«.(«.(«.!>))). («.(«.»)))). c = 2 ' T = THE COMPUTER JOURNAL, VOL. 3, NO. 4,

3 P. SIPALA Notice that, since V N \ =, matrices Q and C have one row and one colmn respectively, that is they are in this case row and colmn vectors, respectively. According to the theory, 5 the matrix A = QC is called the stochastic expectation matrix for the grammar. In general, the element A (j indicates the expected nmber of times the /th non-terminal will occr, when the fth non-terminal is rewritten, sing exactly one prodction. In the present case there is jst one non-terminal S, and A = \\p + q\\. This means that, every time the nonterminal S is rewritten, p + q S"s take its place on the average. A grammar G is called 'consistent' when it generates only strings of a finite length. In this case, to have a 'consistent' grammar, that is, to garantee that, starting with one S, the derivations always terminate, it is clearly necessary to assme that p + q <. The assmption of a 'consistent' grammar will be removed later on in Section 5. According to the theory of probabilistic grammars, 5 the composition of terminal strings is described by the matrix W (terminal expectation matrix), defined as W = (I-A)' Q T (/ is the identity matrix). Here, W=(\-p-q)- -/7 \-q\\. The element W i} of W is the expected nmber of occrrences of the yth terminal in a terminal string derived from therthnon-terminal, as shown in Ref. 5. For example, in the present case the atomic symbol v occrs, on the average, ( q)/(\ p q) times in an S- expression. More information may be obtained by maniplating the matrices Q, C and T, thogh it is not necessary to this argment. 4. REPRESENTATION OF BINARY TREES BY A-ARY CELLS To represent a binary tree by means of A>ary cells, as shown in Fig., consider, within the tree, the seqences of binary cells chained together in the cdr direction. For example, for the tree shown in Fig. 2, look at the three cells seqences formed respectively by the cells {,3,5,7}, {2} and {4,6,8}. The nmber of sch seqences in a tree is clearly eqal to the nmber of v atoms occrring in the S-expression corresponding to the tree; this nmber, as noted in the previos section, has the expected vale To represent the tree by means of k-ary cells, every seqence of cafr-chained binary cells mst be divided into blocks containing k cells each. If the nmber of cells in the seqence is not a mltiple of k, then the first block will contain fewer than k cells. For example, if k = 3, the cell seqences {,3,5,7}, {2} and {4,6,8} of the tree in Fig. 2 mst be divided into blocks of size k \ = 2. The blocks are in this case {,3}, {5,7}, {2}, {4} and {6,8}. If p r is the probability that a cell seqence has length r, then the expected nmber of blocks per seqence is p r \r/{k-\)\ (\x\ denotes the least integer not smaller than x.) The probability that a crfr-chained seqence contains only a single binary cell is q, the probability that it is formed by two binary cells is q{\ q), and so on. In general, p r = ( q)q r ~, becase a seqence of length r contains r list pointers (each occrring with probability q) and one final atom pointer (having probability \ q of occrrence). Therefore the expected nmber of blocks per seqence will be 2 <7 r W-l)lr-l < Becase every block prodces a A>ary cell, containing k pointer fields, and becase there are, on the average ( q)/{\ p q) seqences (or terminal v's) in a tree, it may be conclded that, to represent a binary tree by means of k-ary cells, the reqired nmber of pointer fields is, on the average, c(k,p,q) = \-q It may be easily shown that r-l and therefore c(k,p,q) = k(l-q) This vale indicates the average nmber offields (each containing a pointer or an atom) reqired to represent a binary tree by means of /c-ary cells. It will be convenient in the seqel to se a normalised form of the storage cost c(k,p,q), as a ratio with respect to c(2,p,q), that is, the cost incrred when sing conventional binary cells. Defining c(k,p,q) = c{k,p,q)/c(2,p,q), the previos calclations give k(\ n\ c(k,p,q)= «l l J! (2) It shold be remarked that this reslt has been obtained nder the hypothesis that p + q <, that is, that the probabilistic grammar is 'consistent'. 5. FINITE AND INFINITE TREES Consider now the case p+\ >. This hypothesis, with reference to the probabilistic grammar (), implies that P4 = PI > Pi = ( />) <!) The ineqality means that, in a derivation, it is more likely for the non-terminal 5 to be replaced by two 5"s (prodction P t in ()) than to disappear, generating terminals (prodction Pj). As a conseqence, when p + q >, it is possible, for the probabilistic grammar (), to generate not only S- expressions of finite length, bt also, with a positive probability, infinite ^-expressions, that is infinite binary trees. Moreover, it is clear that the probability of generating finite trees is expressed by the ratio between the probability of sing prodction P and the probability of sing P t. This probability is therefore a = p /p i = (l-p)(\-q)/(pq) <, while -a = (p + q- \)/{pq) is the expected freqency of occrrence of infinite trees. Infinite trees may indeed be interpreted, in LISP, as nfolded representations of circlar list strctres; this will be discssed in the next section. For the present, the sitation for p + q > may be tentatively expressed by means of the following probabilistic grammar (where S generates all trees, F the finite trees and / the infinite ones). Downloaded from by gest on 3 September THE COMPUTER JOURNAL, VOL. 3, NO. 4, 988

4 EFFICIENT IMPLEMENTATION OF BINARY TREES IN LISP SYSTEMS ( («.S); Q, ( ) ; q s pq Q, :/->(«.!>) ; q 7 = Q 9 :I^(S.v) ; q 9 =p(l-q) Q :-* (S.S); q = Pq- -\)/(pq) = (\-p)(\-q) In (3) the prodctions Q and Q 2 express the relative proportions of finite and infinite trees, as discssed above; the prodction sets Q 3 Q s and Q 7 Q, on the other hand, correspond perfectly to those in grammar (), with the exception of Q 7, where q 7 =, becase an infinite tree obviosly cannot terminate. From (3), extract now a sb-grammar for the finite tree poplation, composing the prodctions Q 3 Q e with Q as follows: R :F^(.v) ; r, = q 3 = (l-p)(l-q) R 2 :F- (.F); R 3 :F- (F.v); (F.F); The grammar above is not a 'proper' probabilistic grammar, becase r l + r 2 + r 3 + r i = a = (l-p)(\-q)/(pq)< (a is in fact the overall probability of finite trees). Ignoring for the moment the infinite trees, normalise the grammar above by dividing all probabilities r i by a = ( p)(\ q)/(pq), obtaining in this way qp q(l-p) (\-q)p Comparing grammar (4) with (), they appear to be essentially identical, except for the replacement of p by q (and conseqently of q by p) in the probabilities s t with respect to p t (the initial non-terminal is denoted by F in (4), while in () it is S). It may therefore be conclded that reslt (2), obtained for c(k,p, q) nder the hypothesis p + q <, carries over to the case in which p + q > and only thefinitetrees are taken into accont. The new reslt is obtained from the previos one by a simple replacement, in the formla, of p by q and of q by p. From (2), valid when p + q <, the following expression is derived: holding when p + q > and only finite trees are considered. The two expressions (2) and (5) may be given the nified form c(k,p,q) = 2(i_,*-i)' where (4) ' = min{\-p,q) (6) valid for any vale of/? + q, provided that the contribtion from infinite trees (in the case p + q > ) is ignored. Reslt (6) has been obtained previosly 3 sing a more complicated method, based on the theory of statistical branching processes. 2 Consider valed of the parameters p and q typical of LISP programs, as reported in Refs. 4 and 5, that is, /> =.286, q =.725; then c(k,.286,.725) is minimal for k = 3. Moreover, becase c(3,.286,.725) =.875, the se of three-pointer cells in LISP to represent all (finite) trees may provide a saving of approximately 2% in the storage reqirements with respect to the conventional sage of binary cells. However, the implementation of the basic LISP operators on trees {car, cdr and cons) is somewhat more complex whenfc-arycells are sed than in the binary case. 6. INFINITE TREES IN LISP Considering the vales of p and q typical of LISP programs (p =.286, q =.725), and observing that p + q =. >, it may be conclded that infinite trees may occr in LISP; their freqency, expressed by the probability q 2 = I a = (p + q \)/(pq) in (3), shold be abot./.27, that is abot 5 %. This is in reasonable agreement with the reported data, 3 stating that only a small percentage of the LISP cells are pointed to more than once (those at the head of shared sblists or within cell loops). Infinite trees (in actal practice, circlar list strctres) may arise in LISP in two ways. They may be constrcted by the ser of the langage, by means of field-altering operators, sch as rplaca, rplacd and set} However, in 'pre' LISP, sch operators are not allowed. The only other sorce of circlar strctres in LISP is the constrction, within the system, of'closres' to represent mtally recrsive fnction definitions. 6 These circlar strctres all have a specific shape: they are essentially finite binary trees, except that some of the leaves may have been folded back to point to the root, as shown in Fig. 3. Assme that all infinite trees in LISP have Figre 3. A circlar list strctre in LISP. this particlar strctre (as might be the case for a 'pre' LISP implementation); then it is appropriate to extend to these strctres the previos discssion concerning finite trees. In fact, circlar strctres of this type are nothing else than finite trees, containing, in addition to the terminal symbols and v considered previosly, two other terminal symbols (say ' and v'), indicating a left or right branch folded back towards the root. The general expression (6), therefore, essentially remains the same when taking into accont circlar strctres, that is 'infinite' trees, throgh a simple reformlation of the probabilistic grammar (). 7. IMPLEMENTATION OF LISP OPERATORS ON A-ARY CELLS This section examines the implementation of the basic LISP operators on trees (car, cdr and cons), for the case when trees are represented by linked A>ary cells, of the Downloaded from by gest on 3 September 28 THE COMPUTER JOURNAL, VOL. 3, NO. 4,

5 P. SIPALA type shown in Fig.. Compare Ref. 9 for another implementation, inclding also the field-altering operators rplaca and rplacd, and covering the case of cells of varying size k. A possible approach to the implementation of LISP operators might consist in introdcing selectors and constrctors directly tailored to the A:-ary case. This has been done, for example, in an implementation of LISP, 4 based on the se of both binary (k = 2) and qaternary (k = 4) types of cells. In the latter case, for qaternary cells, the following operators have been introdced: for selectors, car, cbr (= cadr), ccr (= caddr) and cdr (= cdddr), and a constrctor qcons, having k 4 argments. Another possible approach consists in allowing only the conventional LISP operators {car, cdr and cons), which are implemented, however, in terms of k-ary cells. The size k of the cells shold be completely invisible and irrelevant to the programmer. This approach is conceptally otlined in the following program, written for simplicity in Pascal. program kary; const k = 3; {or any other integer > = 2} ' type data = (, v); kind = (list, atom, nll); pkcell = T kcell; fname = record cell: pkcell; index: 2..k field = record case ftype: kind of list: (pointer: fname); atom: (object: data); nll: () kcell = array [..k] of field; The declarations indicate that a k-ary cell (kcell) is an array of k fields; each field may be of one of three different kinds, that is, it may contain a pointer to a list, or it may contain an atom (an object of type data, either the symbol or the symbol v), or,finally,it may be empty (nll). A pointer mst specify, within a k-ary cell, which one of the k \ binary cells packed there it points to. Therefore a pointer shold be of a type (fname) containing both a pointer to the k-ary cell and an index to a position within it. The k \ possible positions are specified by a vale (index) between 2 and k. Procedres car, cdr and cons, implementing the basic LISP operators, may now be declared as follows. procedre car(a:fname; var b:field); {b: = car(a)} begin with a do b: = cell ][index ] procedre cdr(a:fname; var b:field); {b: = cdr(a)} begin with a do if index = k then b: = cew\[k] else begin b.ftype:= list; b.pointer. cell: = cell; b.pointer. index: = index + end The procedre car receives in inpt a pointer and gives as otpt the contents of a field. The field is obtained sing the inpt pointer to get to the relevant A>ary cell and to the relevant position within it, given by index (becase index has been defined to range between 2 and k). The procedre cdr gives as otpt the contents of the link fields between binary cells. These links are actally implied in the k-ary cell representation (see Fig. ), with the exception of the last one (this is the link referred to when index = k). As a conseqence, when index + k, the procedre cdr explicitly constrcts the implied link field and retrns it in otpt. The procedre cons allocates a new binary cell, either within the crrent A>ary cell, if there is room (that is, a nll field) available, or in a new k-ary cell allocated for this prpose. procedre cons(a,b-.field; var c:field); {c:= cons(a,b)} procedre new cell; var d:pkcell; i:..k; begin new(d); d\[k- ]: = a; d][k]: = b; for i:= to k-2 do d][i\. ftype := nll; with c.pointer do begin cell:= d; index: = k end end {allocation of a new k-ary cell}; begin c.ftype: = list; if b.ftype = atom then newcell else with b.pointer do if index = 2 then newcell else if cell\(index -2].ftype =t= nll then newcell else begin cell][index 2]:= a; c.pointer. cell: = cell; c. pointer. index: = index end The procedre cons ses the procedre newcell to allocate a new A>ary cell. In each new cell, all fields are initialised to nll fields, and a pointer is retrned to the rightmost position in the cell. Thefieldsin a k-ary cell are sed by cons, starting from the rightmost one and progressing towards the left. The operator cons constrcts a new binary cell with a car pointer to a and a cdr pointer to b, where a and b are the operands. If there is empty room (that is a nll field) in the A>ary cell, immediately to the left of the b operand, then it is sed for the constrction. If this is not the case a new A>ary cell is allocated, and the constrction is performed within it. To illstrate the sage of the basic operator declarations jst given, consider the following example. procedre printot(a '.field); var /:field; {prints ot the S-expression denoting a tree} begin with a do if ftype = atom then if object = then write(' ') else write(' v') else if ftype = list then begin write('('); car(pointer, t); printot(i); write('.'); cdr(pointer, t); printot(t); write(')') end procedre example; var i, vi, r, s: field; {constrcts the tree of Figre 2 and prints it} begin i.ftype:= atom; i.object:= ; vi.ftype: = atom; vi. object := v; cons(i,vi,r); cons(i,r,r); cons(i,r,r); cons(i,vi,s); cons(i,s,s); cons(r,s,r); cons(i,vi,s); cons(s,r,r) ; printot(r) begin {main program} example end. The program above, when rn, will constrct the data strctre shown in Fig. 4. The tree strctre is explicitly indicated in the procedre example, sing the standard Downloaded from by gest on 3 September THE COMPUTER JOURNAL, VOL. 3, NO. 4, 988

6 EFFICIENT IMPLEMENTATION OF BINARY TREES JN LISP SYSTEMS binary constrctor cons of LISP. However, as an effect of the constant declaration (const k = 3) in the heading of the program, a ternary cell representation is generated internally. It shold be noticed that some of thefieldsin the ternary cells ths constrcted are empty. The representation in Fig. 4 ses altogether 5 fields to based on k-ary cells with different vales of k, are generated, changing the declaration of the constant k. In all cases the procedre printot will print the same S- expression denoting the trees, that is ((.v).((«. (.(.v))).(.(.v)))). The internal representation of the S-expression is irrelevant to the programmer, if only the procedres car, cdr and cons are sed, with no reference to the declarations in the program heading. V Figre 4. Representation of the binary tree in Fig. 2 by means of ternary cells. represent the given S-expression. If we change the constant declaration in the program to const k = 2, then the binary cell strctre of Fig. 2 is prodced internally; this strctre ses 6fields.Corresponding strctres, REFERENCES. J. R. Allen, Anatomy of Lisp. McGraw-Hill, New York (978). 2. K. B. Athreya and P. E. Ney, Branching Processes. Springer, Berlin (972). 3. D. W. Clark, A note on shared list strctre in Lisp. Inf. Process. Lett. 7 (), (978). 4. D. W. Clark, Measrements of dynamic list strctre se in Lisp. IEEE Trans. Software Eng. 5 (), 5-59 (979). 5. D. W. Clark and C. G. Green, An empirical stdy of list strctres in Lisp. Comm. ACM 2 (2), (977). 6. D. P. Friedman and D. S. Wise, Reference conting can manage the circlar environments of mtal recrsion. Inf. Process. Lett. 8 (), 4-45 (979). 7. M. H. Halstead, Elements of Software Science. Elsevier North-Holland, New York (977). 8. D. E. Knth, An empirical stdy of Fortran programs. Softw. Pract. Exper. (2), 5-33 (97). 9. K. Li and P. Hdak, A new list compaction method. Softw. Pract. Exper. 6 (2), (986). V V 8. CONCLUSIONS On the basis of a statistical model for the distribtion of LISP binary trees, sing a probabilistic grammar, ternary cells have been shown to provide a better storage economy than conventional binary ones, both for finite trees and for circlar strctres. The new derivation of this known reslt 3 is a straightforward application of probabilistic grammars. The methods employed can be sed to stdy other programming langage implementation problems; in particlar, they can provide gidance when qestions of optimal implementation strategies arise. Here a detailed implementation has been presented of the basic LISP operators on trees {car, cdr and cons) for the general case of ^-pointer cells.. J. McCarthy, P. W. Abrahams, D. J. Edwards, T. P. Hart and M. I. Levin, Lisp.5 Programmer's Manal. M.I.T. Press, Cambridge, Mass. (965).. R. Sethi, Circlar expressions: elimination of static environments. Science ofcomp. Progr. (3), (982). 2. B. A. Sheil, The psychological stdy of programming. Comp. Srv. 3 (), -2 (98). 3. P. Sipala, Compact storage of binary trees. ACM Trans. Program. Lang. Syst. 4 (3), (982). 4. I. Takechi and H. Okno, A list processor LIPQ. Rev. Electr. Comm. Lab. (Japan) 26 (5-6), (978). 5. C. S. Wetherell, Probabilistic langages. Comp. Srv. 2 (4), (98). 6. G. K. Zipf, Hman Behavior and the Principle of Least Effort. An Introdction to Hman Ecology. Hafner, New York (949). Downloaded from by gest on 3 September 28 THE COMPUTER JOURNAL, VOL. 3, NO. 4,