Remark : Obviously, to each oeig (res closig) bracket there is a uique corresodig closig (res oeig) bracket i every Dyck word w D R The closig bracket

Transcription

1 Rakig ad Urakig of a Geeralized Dyck Laguage Jes Liebeheschel Joha Wolfgag Goethe-Uiversitat, Frakfurt am Mai Fachbereich Iformatik D Frakfurt am Mai, Germay jes@sadsiformatikui-frakfurtde Abstract Give two disjoit alhabets T [ ad T ] ad a relatio R T [ T ], the geeralized Dyck laguage D R over T [ [ T ] cosists of all words w (T [ [ T ] ) which are equivalet to the emty word " uder the cogruece deed by x y " mod for all (x; y) R If the Dyck words are arraged accordig to the lexicograhical order, the rakig meas to determie the rak, i e the ositio, of a Dyck word Urakig creates the Dyck word from its ositio We reset a rakig ad a urakig algorithm for the geeralized Dyck laguage Give a Dyck word, the rakig algorithm reads a rex as short as ossible to comute the rak Thus, this algorithm is otimal The urakig algorithm creates the Dyck word symbol by symbol from left to right Further, we aalyze the rakig algorithm For the comutatio of the rak of a Dyck word for arbitrary R, we comute the s-th momets of the radom variable describig the legth of the rex to be read A aalysis of the urakig algorithm is ot ecessary, because the whole Dyck word has to be created The geeralized Dyck laguage ca be used for the codig of trees Not oly the shae of the tree is coded but also its labels at the odes ad/or edges The algorithms discussed here ca be used for the rakig ad urakig of dieret tyes of trees Further, radom trees ca be geerated with the urakig algorithm Classicatio: algorithms ad data structures, comutatioal comlexity Keywords: Dyck laguage, rakig, urakig, lexicograhical order, trees, oe-to-oe corresodece, average-case aalysis Itroductio ad Basic Deitios I this aer we examie the geeralized Dyck laguage that was itroduced i [Ke96b] A orderig o the alhabet will be cotiued to a orderig o the Dyck words, that is called lexicograhical order I [Li98], a algorithm for the geeratio of all Dyck words accordig to their lexicograhical order was reseted Give a set of ordered words, every word has a ositio, that is called rak; the comutatio of the rak is called rakig Urakig is the iverse oeratio, i e the geeratio of a word from its ositio First, we itroduce the geeralized Dyck laguage ad the lexicograhical order, the we give a short examle I the ext sectio, the rakig ad urakig algorithms are reseted I Sectio 3, the rakig algorithm is aalyzed Deitio : Let t, t N ad T [ := [ ; [ ; : : : ; [ t res T] := ] ; ] ; : : : ; ] t be the set of oeig (res closig) brackets Let jsj be the cardiality of the set S, so = t ad T] = t With T := T [ [ T ] ad a relatio R T [ T ] we obtai the geeralized Dyck laguage associated with R by D R := fw T j w " mod g, where " deotes the emty word ad is the cogruece over T which is deed by (8([ a ; ] b ) R)([ a ] b " mod ) The set of all Dyck words of legth is give by D R := D R \ T

2 Remark : Obviously, to each oeig (res closig) bracket there is a uique corresodig closig (res oeig) bracket i every Dyck word w D R The closig bracket corresodig to a oeig bracket i a Dyck word w D R ca be foud by searchig for the rst closig bracket behid the shortest word w D R (w might be ") o the right side of the oeig bracket If the oeig (res its corresodig closig) bracket is [ a (res ] b ), we have w = w [ a w ] b w 3 with w w 3 ; w D R The oeig bracket corresodig to a closig bracket i a Dyck word ca be foud i a aalogous way Remark : It is well-kow that there are several oe-to-oe corresodeces betwee the ormal Dyck laguage ad other combiatorial objects For examle, the shae of a ordered tree or a exteded ordered biary tree ca be coded i the ormal Dyck laguage (for examle [Za80]), but the labels at the odes or the edges ca ot be coded i the ormal Dyck laguage I [FS96, 66] was metioed a codig of ordered trees with labelled odes by aother geeralizatio of the Dyck laguage With the geeralizatio discussed here, we ca also code the labels at the edges Additioally, coectios betwee the labels at the root of a subtree ad the edge to its father (i a ordered tree) ad coectios betwee the labels at the left ad right sos ad/or edges (i a exteded ordered biary tree) ca be cosidered Some ossibilities of how to code a tree i a Dyck word are give i Figure b a b a Y a Y b a b Y Figure : Corresodig trees to the Dyck word [ a ] b Y The oe-to-oe corresodeces betwee the trees ad the Dyck words are deed recursively All trees i Figure corresod to the Dyck word [ a ] b Y, where [ a ad ] b are corresodig brackets ad ; Y D R The left icture shows a ordered tree, whereas the other ictures rereset exteded ordered biary trees I the right icture, a ad a (res b ad b ) must be coded i a (res b) Note that i each case the label at the root of the tree is ot coded i the Dyck word It ca be coded { if eeded { i oe more symbol to the left or right of the Dyck word Remark 3: With the urakig algorithm we have a rocedure at had, that geerates radom trees of various kids (see Remark ) We use a radom umber geerator ad comute the Dyck word ad the corresodig tree, whose rak is equal to the radom umber Deitio : Let close R ([ a ) := f ] b j ([ a ; ] b ) Rg be the set of all closig brackets corresodig to a give oeig bracket accordig to the relatio R

3 Deitio 3: Let < lex T T be a irreexive liear orderig o T The lexicograhical order lex over T + is deed as the extesio of < lex to lex T + T + by x lex y :, (9z T + ) (xz = y) _ 9 (u; ex; ey; v;ev) T 3 T (x = u v ex ^ y = uev ey ^ v < lex ev) [Ke98] Note that i this aer we cosider the lexicograhical order o Dyck words of legth oly We use the followig orderig o T : [ j j < lex : : : < lex [ < lex ] < lex : : : < lex ] jt] j If all Dyck words of D R are arraged accordig to the lexicograhical order, the rakig meas to determie the rak, i e the ositio of a Dyck word w D R The rak of w is the umber of lexicograhically smaller Dyck words i the Dyck laguage: rak(w) := few j ew D R ^ ew lex wg, thus 0 rak(w) < jd R Give the rak of w D R, urakig comutes w Let us demostrate the above deitios by a simle examle Examle : Let T := f[ ; [ ; ] ; ] g, R := f([ ; ] ); ([ ; ] ); ([ ; ] )g ad := The relatio R imlies close R ([ ) = f ] ; ] g ad close R ([ ) = f ] g We obtai the orderig o the alhabet [ < lex [ < lex ] < lex ] I Figure all Dyck words of this examle are arraged lexicograhically Cosiderig the rak of the Dyck words, we d rak([ [ ] ] ) = 0, rak([ [ ] ] ) = 0 ad rak([ ] [ ] ) = 7 = jd R 4 j, for examle j [ [ ] ] lex [ [ ] ] lex [ [ ] ] lex [ ] [ ] lex [ ] [ ] lex [ ] [ ] lex [ [ ] ] lex [ [ ] ] lex [ [ ] ] lex [ [ ] ] lex [ [ ] ] lex [ [ ] ] lex [ ] [ ] lex [ ] [ ] lex [ ] [ ] lex [ ] [ ] lex [ ] [ ] lex [ ] [ ] Figure : All Dyck words of Examle arraged accordig to the lexicograhical order lex Rakig ad Urakig Algorithm I this sectio we will reset a rakig algorithm for the geeralized Dyck laguage D R as well as a urakig algorithm, that creates the Dyck word by reversig the stes carried out by the rakig algorithm Sice a Dyck word's rak is equal to the umber of lexicograhically smaller Dyck words, the rakig algorithm eeds to determie that umber It follows the familiar ractice ad reads the Dyck word from left to right As show i the geeral aroach of rakig ad urakig [Li97], the algorithm has to go o readig util there is exactly oe cotiuatio to a word of the laguage I that case the symbols ot read so far are uiquely determied by the symbols read Give a Dyck word w = w 0 : : : w, we comute the umber of lexicograhically smaller words by rak(w) := where A i := ew 0 : : : ew D R j ew 0 : : : ew i = w 0 : : : w i ^ ew i < lex w i Note that A i is the umber of Dyck words with the same rex of legth i as w ad a smaller symbol at the right of that commo rex tha i w Furthermore, each A i, 0 i, ca be comuted by readig the rst i symbols of w, so that the algorithm ca comute the rak by readig the Dyck word from left to right i=0 A i, 3

4 If the shortest rex with a uique cotiuatio is read, the algorithm reseted here does ot read aother symbol So, it is otimal with resect to the umber of symbols to be read from left to right Let j be the legth of that shortest rex with a uique cotiuatio We immediately d 0 j Obviously, our algorithm is ot able to comute A i, j i, because it does ot read the sux w j : : : w of w The comutatio of A i, j i, is ot eeded, because that sux w j : : : w is a uique cotiuatio of the rex read to a Dyck word i D, R thus we get A i = 0, j i, by the deitio of A i Now, we have the fudametals for the comutatio of the rak of a Dyck word w The algorithm roceeds as follows: It reads w from left to right For each ositio i, 0 i, is checked, whether exactly oe cotiuatio to a Dyck word of legth exists That is the case, i either all oeig brackets have bee read ad each of these corresods to oe closig bracket oly or ( ) oeig ad ( ) closig brackets have bee read ad the relatio cotais oly oe air of brackets If such a uique cotiuatio exists, the o more symbols have to be read, as we have see before If ot, the the algorithm has to comute A i, whereby it is ecessary to distiguish betwee oeig ad closig brackets: (i) For a oeig bracket, the umber of Dyck words with the same rex as w ad a smaller oeig bracket at the curret ositio tha i w have to be added to the rak (ii) For a closig bracket, the umber of Dyck words with the same rex as w ad a oeig bracket at the curret ositio have to be added to the rak as well as the umber of Dyck words with a smaller closig bracket tha i w { i cosideratio of the corresodig oeig bracket Thus, it is ecessary to kow the oeig bracket corresodig to each closig bracket For that urose the algorithms make use of a stack with the oeratios ush ad o Furthermore, the algorithms eed to comute the umber of cotiuatios of a Dyck word's rex to a Dyck word of legth Therefore, let us recall the well-kow oe-to-oe corresodece betwee Dyck words ad aths i the lattice give i Figure 3 j (;) (i;j) i Figure 3: Oe-to-oe corresodece betwee Dyck words of legth [ a ad the aths from (0; 0) to (; 0) ] b (i+;j+) (i+;j) Thereby, a u-segmet % (res dow-segmet &) corresods to a oeig (res closig) bracket Sice we have a corresodece betwee oeig ad closig brackets (see Remark ), there must be a corresodece betwee u-segmets 4

5 ad dow-segmets i the lattice To each u-segmet there is a corresodig dow-segmet ad vice versa: The dow-segmet (res u-segmet) corresodig to a u-segmet (res dow-segmet) is the ext segmet o the right (res left) side i the same row Thus, each Dyck word of legth corresods to a ath from (0; 0) to (; 0) The segmets are labelled by the symbols of the alhabet; the labels of u-segmets (res dow-segmets) are oeig (res closig) brackets Obviously, a usegmet ad its corresodig dow-segmet must be labelled by some [ a ad ] b with [ a ; ] b R Note that the umber of aths from (0; 0) to (i; j) i the lattice is give by the ballot umber [CRS7] (i; j) = j + i + i + (i + j) + = i (i j) i (i j) () Now, the umber of cotiuatios ca be comuted by ad two other factors, oe for the umber of ossibilities for the closig brackets ot read so far ad oe for the airs of brackets ot read so far The umber of aths from (i; j) to (; 0) multilied by those factors described above is give by: q(i; j) := ( i; j) close jrj ij, where close is a variable i the algorithms The variables used i the rakig algorithm will be exlaied ow u : Number of oeig brackets (ad u-segmets) dow : Number of closig brackets (ad dow-segmets) Q close := k j=0 close R (ew j ), if w0 : : : w i ew 0 : : : ew k mod ^ w 0 : : : w i is read i the algorithm ^ ew 0 : : : ew k T [ k Note that close is the roduct of the umbers of closig brackets i R corresodig to each oeig bracket read without its corresodig closig bracket read so far, i e umber of ossibilities to comlete the rex read to a Dyck word i D R by aedig closig brackets oly This variable is eeded for the fuctio q ad will be recomuted for each symbol beig read i both algorithms corr cl br[a ] := R close ([ a ), a j j P corr cl br sm br[a ] := a R close ([ l ), a j j l= sm corr cl br[a ][b ] := fx j x close R ([ a ) ^ x < lex ] b g, a j j, b jt ] j Note that the arrays corr cl br ad corr cl br sm br ad the matrix sm corr cl br ca be recomuted i O(jT [ j jt ] j) This amout of time is costat with resect to the legth of the words, because the variables deed o the relatio R oly We code the brackets i itegers i the rakig ad urakig algorithms; the oeig bracket of tye a (res closig bracket of tye b) is coded i the iteger a (res b) Note that w = w[0] : : : w[] is a array of itegers i the algorithm The formal rakig algorithm is give as follows 5

6 fuctio rak (w : dyck word) begi rak := 0 u := 0 dow := 0 close := i := 0 while i < do begi = check if a uique cotiuatio exists = if u = ad close = or jrj = ad u = ad dow = the = do ot cotiue readig the Dyck word = i := else begi if w[i] < 0 the begi = w[i] is a oeig bracket = rak := rak + corr cl br sm br[w[i] ] q(i + ; i + ( dow)) ush(w[i]) close := close corr cl br[w[i] ] u := u + ed else begi = w[i] is a closig bracket = rak := rak + jrj q(i + ; i + ( dow)) ed; ed j := o() close close := corr cl br[j] rak := rak + sm corr cl br[j ][w[i] ] q(i + ; i ( dow)) dow := dow + ed i := i + ed Algorithm : Rakig algorithm for the geeralized Dyck laguage D R The urakig algorithm reverses the stes made by the rakig algorithm It creates the word successively from left to right The algorithm has to be determie for each ositio, whether the curret symbol is either a oeig or a closig bracket The, the tye of that bracket has to be comuted Further, a recomutatio of the rak is ecessary; this corresods to the stewise comutatio of the rak (A i, 0 i ) i the rakig algorithm To comutate of the tye of a oeig or closig bracket, two more variables are eeded by the urakig algorithm: o br[l ] : oeig bracket of the l-th air of brackets of the relatio R accordig to the lexicograhical order o the airs of brackets, l jrj cl br[a][l ] : l-th closig bracket that corresods to [ a i the relatio R accordig to the orderig o the alhabet, a jt [ j, l close R ([ a ) 6

7 Agai, the array o br ad the matrix cl br ca be recomuted i O(jT [ j jt ] j) Now, let us have a look at the formal urakig algorithm fuctio urak (rak : iteger) begi dow := 0 close := i := 0 while i < do begi k := q(i + ; i + ( dow)) if rak < jrj k the begi = urak[i] is a oeig bracket = urak[i] := o br[b rak c] k rak := rak corr cl br sm br[urak[i] ] k ush(urak[i]) close := close corr cl br[urak[i] ] ed else begi = urak[i] is a closig bracket = rak := rak jrj k j := o() ed; close := close corr cl br[j] k := q(i + ; i ( dow)) urak[i] := cl br[j ][b rak c] k rak := rak sm corr cl br[j ][urak[i] ] k dow := dow + ed i := i + ed Algorithm : Urakig algorithm for the geeralized Dyck laguage D R 3 Aalysis of the Rakig Algorithm I this sectio we aalyze the average umber of symbols to be read by our rakig algorithm The aalysis of the urakig algorithm is ot ecessary, because the algorithm always has to create the whole Dyck word, so the umber of symbols to create is equal to for every w D R Thus, the mea value is equal to ad the variace is equal to 0 Sice the rakig algorithm reads the shortest rex, that is log eough to comute the rak of the Dyck word, we ca aalyze the umber of symbols to be read usig the geeral aroach to rakig ad urakig [Li97] Let ref (D R ) be the radom variable describig the legth of the shortest rex to be read i order to rak a Dyck word The mea value ref (D R ) ad the variace ref (DR ) of ref (D R ) are give by: ref (D R ) = E[ ref (D R )] ad ref(d R ) = E[ ref (D R )] E[ ref (D R )] 7

8 We obtai the s-th momets, s, of ref (D R ) by [Li97]: where E[ s ref (D R )] := () s D R k= ((k + ) s k s )! INIT k(d) R + jd R j; INIT k(d R ) := u j u v D R ^ u T k ^ y D R j y = u T k =, () is the set of rexes of legth k with a uique cotiuatio to a word belogig to D R ad is Kroecker's delta Agai, we eed a criterio for a uique cotiuatio of a Dyck word's rex Let us formalize the coditio for a uique cotiuatio that has bee give i Sectio Regardig the lattice i Figure 3, we see that a uique cotiuatio to (; 0) exists from all oits of f( + l; l) j 0 l g, if the labels of all dowsegmets are kow; i other words all oeig brackets are read ad the tyes of the closig brackets ot read so far ca be determied with kowledge of the rex read Moreover, a uique cotiuatio exists from ( ; 0), if the labels of the last two segmets are kow, i e if oly oe air of brackets is ot read so far ad jrj = We ca comute the cardiality of INIT k(d R ), k <, ow: INIT = k(d) R 8 >< >: 0 if k (k; k) k jrj k if k 3 ( ; ) + ( ; 0) if k = ^ jrj = ( ; ) jrj if k = ^ jrj ( ; ) jrj if k = (3) Here, T [ := fx j x T [ ^ jclose R (x)j = g is the set of oeig brackets with oly oe corresodig closig bracket accordig to R ad is give i () Before comutig the s-th momets of ref (D), R we roof the followig techical lemma which gives us the asymtotic behaviour for two tyes of sums Lemma : The asymtotics for both sums are valid for! (i) The followig asymtotic [Ke96a] holds for s N 0 ad > : ( + ) s k + = (4) 3 () s 4 ( ) ( + O( )) (4) (ii) The followig asymtotic holds for > : ( k) k + = (4) 3 8 ( ) 3 ( + O( )) (5) 8

9 Proof: To comute the asymtotic equivalet for the sum i (ii) we rst cosider the followig geeratig fuctio for c N 0 G(z; c; ) := z ( k) k + c 0 = 0 z k k k + c After rearragig the sums ad shiftig the idex of summatio, we get: With the idetity l + l0 + + c G(z; c; ) = z (z) +c 0 0 [GKP94, 03], we ca rewrite (6): l (z) l 4z =, N 0, 4z z 4z G(z; c; ) = (z) c 4z z c 0 (6) 4z Now, we cosider the derivative of the geometric series of a dieretiable fuctio f(z) with f 0 (z) 6= 0 d dz 0 f(z) = d dz f(z) ; 0 With that idetity, we immediately get G(z; c; ) = 4z z c 4z f(z) = ( f(z)) c+ 4z (7) Now, with (7) for c = 0 ad c = we d the geeratig fuctio of the sum i (ii) Note that the term for k = i the sum with idex of summatio k is equal to 0 We obtai the simle closed form of the geeratig fuctio H(z; ) by a straight forward comutatio H(z; ) := 0 = 4z z ( k) k 4z z 4z = z 4z + " 4z z 4z Now, we are lookig for the sigularity of smallest modulus that is ot equal to 0 We d ossible sigularities at z = 0, z = ad z = We do ot have to 4 take z = 0 ito accout Further, z = is o sigularity for >, because H( ; ) = ( ) 4( ) ( ) 4( ) = ( ), > 9 #

10 Thus, the domiat sigularity is at z = Now, alyig Darboux's method 4 [Ke84], we get the asymtotic behaviour of the sum stated i the lemma With this lemma we are able to aalyze the radom variable ref (D) R Theorem : Assumig that all words i D R are equally likely, the s-th momets, s, of the radom variable ref (D) R are give by: Proof: With the abbreviatio := jrj the idex of summatio E[ s ref (D R )] = E[ s ref (D R )] = () s + O( s ) jt [ j () s D R we get by () usig (3) ad () ad by shiftig ( ( + ) s k + ) () s k + {z } =:f (;) The asymtotic behaviour of the rst sum is give by (4) To gai a asymtotic equivalet for the secod sum f(; ), we rst aly the additio formula for a biomial coeciets b = a b + a b We d f(; ) = g(; ) + h(; ), (8) where g(; ) := h(; ) := () s k () s k +, By shiftig the idex of summatio ad alyig obtai g(; ) = ( + ) s k+ + () s + + = +, we By (4) ad the asymtotic behaviour of the Catala umbers comuted by Stirlig's formula [GKP94] = 4 3 ( + O( )) (9) + we immediately get: g(; ) = (4) 3 () s 3 ( ) ( + O( )) 0

11 A similar comutatio yields h( + ; ) = ( + ) s k = (4) 3 + ( + ) s + + () s 4 ( ) + ( + )s ( + O( )) Now, usig the asymtotics for g(; ) ad h(; ) i (8) ad alyig the formula for the umber of Dyck words jd + R j = jrj [Li98], a simle comutatio leads to the equatio E[ s (DR ref )] = () s ( + O( )) With E[ s (DR ref )] we are ot able to comute ref (D) R ad (DR ref ) with recisio O( ) This will be doe i Theorem ad Theorem 3 Obviously, (3) imlies that we must distiguish betwee the cases jrj = ad jrj The case jrj = has bee ivestigated i [Li97] There we d the followig theorem Theorem : Assumig that all words i R are equally likely, the mea value ad the variace of the radom variable ref (D) R for jrj = ad are give by: (i) ref (D R ) = O( ), (ii) ref(d ) R = O( ) For the case jrj we get the mea value ad the variace of the umber of symbols to be read i the followig theorem Theorem 3: Assumig that all words i R are equally likely, the mea value ad the variace of the radom variable ref (D) R for jrj are give by: Case : = 0, (i) ref (D R ) =, (ii) ref(d R ) = 0 Case :,! (i) ref (D R ) = 4 ( ) + O( ), (ii) ref(d R ) = 4 (4 3) ( ) 4 + O( ) Agai, = jrj jt [ j Proof: The roof of Case is trivial We isert (3) i () ad obtai E[ ref (DR )] = ad E[ ref (DR )] = () For the mea value i Case we get: ref (D) R = E[ (D R ref )] = jdj R (k; k) k jrj k k= = jd R j (; k) jrj! k

12 With the abbreviatio = jrj jt [ j, () ad jdr j = + jrj [Li98], we obtai ref (D) R = + k + By (4) ad (9) we immediately get the asymtotics for ref (D R ) For the variace, we obtai with (), (3) ad () ref(d R ) = E[ ref (D R )] E[ ref (D R )] = () jd R j ( + k + ) jd R j Simlifyig this exressio yields ref(d R ) = jd R j ( k) jd R j jd R j jrj! k! + + jrj! jrj k! jrj! k! k jrj! k! By alyig (4), (5) ad (9) ad erformig simlicatios, we directly get the variace stated i the theorem Remark 4: I [Li97] the rakig ad urakig of the Dyck laguage with t tyes of brackets D t [Ha78, 33] was aalyzed The formal deitio of D t is as follows: Let t N ad T := f[ ; ] ; : : : ; [ t ; ] t g be the alhabet with the liear orderig [ t < lex : : : < lex [ < lex ] < lex : : : < lex ] t The Dyck laguage D t is the smallest subset of T satisfyig (i) ad (ii): (i) " D t, (ii) u; v D t ; [ u ] v D t ^ : : : ^ [ t u ] t v D t The Dyck laguage with airs of balaced brackets ad t tyes of brackets is give by D t := D t \ T Note that D t = D R with the relatio R = f([ ; ] ); : : : ; ([ t ; ] t )g The mea value ad the variace of the radom variable ref (D t ) describig the umber of symbols to be read for comutig the rak for t ^ was foud to be: (i) ref (D t ) = 3 + O( ), (ii) ref(d t ) = 4 + O( ) With R = f([ ; ] ); : : : ; ([ t ; ] t )g, t, we d jrj = = t This imlies = jrj = Thus, we obtai by Theorem 3, Case, the results stated i [Li97] jt [ j

13 4 Cocludig Remarks I this aer we reseted a rakig ad a urakig algorithm for the geeralized Dyck laguage The Dyck laguage is deed by a relatio R which describes the airs of brackets that ca be used Followig a geeral aroach to rakig ad urakig [Li97], we comuted the s-th momets of the radom variable describig the legth of the rex to be read i order to determie the rak of a Dyck word Esecially, we oited out the mea value ad the variace of the legth of that rex with recisio O( ) for the Dyck laguage D R We showed that the umber of symbols to be read is liear i the legth of the words with a costat variace The urakig algorithm always has to create the whole word, so the mea value of the umber of symbols to be created is equal to the legth of the words ad the variace is equal to 0 Both algorithms reseted here are otimal with resect to the umber of symbols to be read (res created) from left to right Further, we metioed oe-to-oe corresodeces betwee the geeralized Dyck laguage ad ordered as well as exteded ordered biary trees With the geeralizatio of the Dyck laguage we are able to code ot oly the shae of a tree but also the iformatio at the odes ad/or the edges Thus, with the algorithms reseted here, we have rakig ad urakig algorithms for a liear codig of these two kids of trees at had Refereces [CRS7] [FS96] L Carlitz, D P Roselle ad R A Scoville, Some Remarks o Ballot-Tye Sequeces of Positive Itegers, Joural of Combiatorial Theory, 97, 58-7 P Flajolet ad R Sedgewick, A Itroductio to the Aalysis of Algorithms, Addiso-Wesley, 996 [GKP94] R L Graham, D E Kuth ad O Patashik, Cocrete Mathematics, d ed, Addiso-Wesley, 994 [Ha78] [Ke84] [Ke96a] [Ke96b] [Ke98] [Li97] M A Harriso, Itroductio to Formal Laguage Theory, Addiso- Wesley, 978 R Kem, Fudametals of the Average Case Aalysis of Particular Algorithms, Wiley-Teuber, 984 R Kem, O Prexes of Formal Laguages ad their Relatio to the Average-Case Comlexity of the Membershi Problem, Joural of Automata, Laguages ad Combiatorics, 997, 4, R Kem, O the Average Miimal Prex-Legth of the Geeralized Semi-Dycklaguage, RAIRO Theoretical Iformatics ad Alicatios 30, 996, 6, R Kem, Geeratig Words Lexicograhically: A Average-Case Aalysis, Acta Iformatica 35, 998, 7-89 J Liebeheschel, Rakig ad Urakig of Lexicograhically Ordered Words: A Average-Case Aalysis, Joural of Automata, Laguages ad Combiatorics, 998, 4,

14 [Li98] [Za80] J Liebeheschel, Lexicograhical Geeratio of a Geeralized Dyck Laguage, Techical Reort: Iterer Bericht 5/98, Fachbereich Iformatik, Joha Wolfgag Goethe-Uiversitat, Frakfurt am Mai, Germay, submitted S Zaks, Lexicograhic Geeratio of Ordered Trees, Theoretical Comuter Sciece 0, 980,