2 A bijective proof of Theorem 1 I order to give a bijective proof of Theorem 1, we eed to itroduce the local biary search (LBS) trees, used by Postio

Transcription

1 Eumeratig alteratig trees Cedric Chauve 1, Serge Dulucq 2 ad Adrew Rechitzer 2? 1 LaBRI, Uiversit Bordeaux I 31 Cours de la Lib ratio, 3340 Talece, Frace 2 Departmet of Mathematics, The Uiversity of Melboure Parville, Victoria 302, Australia?1, Abstract. I this paper we examie the eumeratio of alteratig trees. We give a bijective proof of the fact that the umber of alteratig urooted trees with vertices is give by 1 P 2?1 =1? a problem rst posed by Postiov i [4]. We also prove, usig formal argumets, that the umber of alteratig plae trees with vertices is 2(? 1)?1. 1 Itroductio ad mai results Deitio 1. A tree T o the set of vertices [] = f1; 2; : : : ; g is said to be alteratig if for every path i 1 ; i 2 ; i 3 ; i 4 ; : : : i T we have i 1 < i 2 > i 3 < i 4 > : : : or i 1 > i 2 < i 3 > i 4 < : : :. Remar 1. I this paper, uless specied, we cosider trees havig a distiguished vertex, called the root. I a recet wor [4], Postiov gives a formula for eumeratig urooted alteratig trees ad a fuctioal equatio satised by their geeratig fuctio. These results are summarised i the ext two theorems. Theorem 1. [4] Let F be the umber of urooted alteratig trees o []. The F = 1 2?1 =1?1 : Theorem 2. [4] Let F (t) be the shifted geeratig fuctio for the umber of urooted alteratig trees o [], t F +1 F (t) = 0 The F (t) satises the followig fuctioal equatio: F = e t 2 (F +1) : To prove Theorem 1, Postiov rst proves Theorem 2 ad the solves the equatio usig the Lagrage's Iversio Formula. I Sectio 2 we preset a bijective proof of Theorem 1, which aswers a problem posed by Postiov. I Sectio 3, we cosider the family of alteratig plae trees (trees such that the sos of each vertex are ordered) ad we prove the followig result. Theorem 3. Let G be the umber of alteratig plae trees o []. The : G = 2(? 1)?1 : Corollary 1. Let H be the umber of urooted alteratig plae trees o []. The H = (? 1)?2 :? [chauve,dulucq]@labri.u-bordeaux.fr ad adrewr@ms.uimelb.edu.au

2 2 A bijective proof of Theorem 1 I order to give a bijective proof of Theorem 1, we eed to itroduce the local biary search (LBS) trees, used by Postiov i [4]. Deitio 2. A LBS tree is a labeled biary plae tree such that every left so has a smaller label tha its paret, ad every right so has a larger label tha its paret. For example, the followig tree is a LBS tree with 9 vertices Theorem 4. [4] The umber of LBS trees o [] such that the root has oly oe so is F = 1 2?1 =1?1 : To prove this result, Postiov gives a bijectio, called, betwee LBS trees o [] such that the root has oly oe so ad alteratig trees o [] (there are F such trees) ad uses Theorem 1. Now, we give a bijective proof of Theorem 4, which, combied with the bijectio of Postiov, provides a complete bijective proof of Theorem 1. I fact we describe a bijectio betwee two families of trees, B ad T, that we dee below. We deote by B the set of LBS trees o [] such that the root has oly oe so ad every oroot vertex is mared with a letter chose from fy; Ng. It follows immediately from the bijectio that jb j = 2?1 F : (1) We deote by T the set of trees o [] such that every leaf is mared by a letter chose from fy; Ng. For a tree T of T, we deote by it(t ), leaves Y (T ) ad leaves N (T ) respectively its umber of iteral vertices, its umber of leaves mared with Y ad its umber of leaves mared with N. From the Pr fer ecodig of trees (see [] or [3]), we ca say that the trees of T with it(t ) + leaves Y (T ) = (1 ) are i bijectio with the set of pairs (A; w) where A [], jaj = ad w is a word of legth? 1 o the alphabet A. Therefore, the umber of trees of T such that it(t ) + leaves Y (T ) = is ad jt j = =1?1 ;?1 : (2) The rst step i the descriptio of the bijectio is a ivolutio o LBS trees of B. Let T be a LBS tree o the set of vertices fx 1 ; x 2 ; : : : ; x g, with x 1 < x 2 < : : : < x. The LBS tree (T ) is obtaied by performig the followig operatios:

3 swap labels x i ad x?i+1 for 1 i, for every vertex, swap its right subtree with its left subtree, the mars o the leaves stay attached to the vertices ad ot to the labels. 3 N 6 Y 7 N 2 Y 1 N 2 N 9 N 8 Y 7 Y 4 N 9 Y 1 Y 6 N 3 Y We ca ow give a recursive descriptio of the bijectio betwee T ad B. We suppose that the iput is a tree T of T ad we wat as output a tree (T ) = B of B. 1. If T has oly oe vertex (it is a leaf mared with a letter C of fy; Ng), the B = T. 2. If T has two vertices, the we distiguish two cases: if T is the tree with root labeled 1 ad a leaf labeled 2 ad mared with C 2 fy; Ng, the B is the LBS tree of root labeled 1 with a right so labeled 2 ad mared with C, otherwise, T has root labeled 2 ad a leaf labeled 1, mared with C, ad B is the LBS tree havig root labeled 2 with a left so labeled 1 ad mared with C. Remar 2. From these rst two cases, we ca use the hypothesis that the root of (T ) has the same label tha the root of T ad has at most oe so. 3. I the geeral case, T has at least 3 vertices. Let r be the root, x 1 ; x 2 ; : : : ; x its sos ad T 1 ; T 2 ; : : : ; T the subtrees of T with respective roots x 1 ; x 2 ; : : : ; x. For each tree T i, we deote its image (T i ) = B i (usig Remar 2, we ca suppose that the root of B i is x i ad has at most oe so) ad we dee the tree B 0 i by the followig rules: if B i has oly oe mared vertex x i, the B 0 i = B i, if x i has oly oe right so, the B 0 is i B i with the root mared N, if x i has oly oe left so, the B 0 i = (B i), with its root (that ca dier from x i ) mared with Y (so the root of B 0 i has oly a right so). Now we have a set of LBS trees such that each vertex is mared with a letter of fy; Ng. We deote y 1 ; y 2 ; : : : ; y their roots such that y 1 < y 2 < : : : < y, ad we ed the costructio of B: if y < r, the set y to be the left so of r, y?1 the left so of y, etc, if y > r, the set y to be the right so of r, y?1 the left so of y, y?2 the left so of y?1, etc. Remar 3. It is immediate to chec the validity of the hypothesis about the root of (T ) give i Remar 2. Now, we give a example. Let T be the followig tree of T Y 7 2 Y 9 6 N 3 Y T 3 1 N T 2 T 1 T 4

4 The we have B 0 N B 0 1 = B 1 = 1 N B 0 2 = B 2 = 3 Y B 3 = B 0 3 = B 4 = 7 4 = 6 Y 7 Y 6 Y 2 Y 9 N 2 N 9 Y ad ally, (T ) is the followig tree. 4 6 Y N 7 Y 3 Y 9 Y 1 N 2 N The costructio of the iverse map is clear ad we have a bijective proof of Theorem 1. To coclude this sectio, we give some additioal eumerative results o the trees cosidered here. Corollary 2. Let B ;p be the set of LBS trees o [] such that the root has oly oe so u ad the left brach from u (the path issued from u ad followig oly left edges) has p vertices. jb ;p j = 2?1? 2 p? 1?1 =0? 1?1?p : Proof. From the deitio of, we ca say that 2?1 jb ;p j is the umber of trees o [] such that the root has exactly p sos ad every leaf has a mar chose from fy; Ng. The result follows immediately from the Pr fer ecodig of such trees ad a easy computatio. 2 Corollary 3. The umber of trees o [] such that every vertex has at least a so greater tha it (i other words, for every vertex u there is a icreasig path from u to a leaf) is 1 F = =1?1 : Proof. This result is a cosequece of the limitatio of o LBS trees such that the root has oly a right so ad every vertex has the mar N. I this way, we recogize the classical correspodece betwee biary trees ad trees (see [2]) limited to LBS trees. 2 3 Eumeratig alteratig plae trees I order to prove Theorem 3, we will focus o the family of alteratig plae trees such that the root is lower tha all of its sos. Let L deote the umber of such trees o []. We will show that L = (? 1)?1, ad the deduce immediately Theorem 3. We deote L(T ) the expoetial geeratig fuctio of such trees: L(t) = 1 L t :

5 Lemma 1. The geeratig fuctio L satises the fuctioal equatio L? 1 =?e t L?1 : Proof. We have clearly the followig relatio betwee G ad L : G 1 = L 1 = 1; G = 2L ( > 1): (3) Now, cosider the umber of dieret plae trees o vertices we ca geerate from a urooted plae tree T o the same set of vertices. For each edge (u; v) of T, it cosists to decide which of the vertices u ad v will be the root of a ew tree, the other beig its leftmost so. The we have the followig relatios betwee G ad H ad betwee L ad H G 1 = H 1 = 1; G = 2(? 1)H ( > 1); (4) L 1 = H 1 = 1; L = (? 1)H ( > 1): () Fially, we itroduce the expoetial geeratig fuctio H(t) H(t) = 0 H +1 t Usig classical results o expoetial geeratig fuctios ad labeled structures (see [1]), we ca say that: H = 1 + l 1 ; (6) 1? L ad the Now, the rest of the proof is calculus usig the previous relatios. t L = t t + (? 1)H = 1? : (7) 2 =) L = th? 2 H t By dieretiatio ad usig equatio (7), we = 1 L? 1? L? =) 1 L? (L? 1) 2 )? 1 L? 1 Itegratig the previous expressio withl(0) = 0, we have t = l(?1)? l(l? 1) = 0:

6 =) L? 1 =?e t L?1 : Corollary 4. Let T (t) be the expoetial geeratig fuctio for o-empty trees o []: T (t) = 1?1 t : The, the geeratig fuctios T (t) ad L(t) verify the relatio L? 1 =? 1 e T : Proof. It is well ow that (see for example [6] or [3]) From Lemma 1, we have: L? 1 =?e L?1 t =)? t L? 1 = t t e? L?1 : Ad the, from equatio (8), we ca say that: T = t e T (8)? t L? 1 = T =) T = t + L T =) L? 1 =? 1 e T : 2 I order to perform the last step of the proof of Theorem 3, we eed the followig versio of the Lagrage iversio formula (see [1, page 6]). P Theorem. (Lagrage Iversio) Let f (z) = i0 f iz i be a formal power series with f 0 6= 0, ad let Y (t) be the uique formal power series solutio of the equatio Y = tf (Y ). The the coeciets of g(y ) (for a arbitrary series g) are give by [t ]g(y (t)) = 1 [z?1 (z): We ca ow ed the proof of Theorem 3. It suces to apply the previous result to the equatio give i Corollary 4, with the correspodeces f (t) = e t ad g(t) =?e?t. The, we have [t ](L(t)? 1) = 1 [z?1 ]e (?1)z = [z?1 ] P 0 (?1) z = (?1)?1 ; which eds the proof of Theorem 3. The proof of Corollary 1 is a direct cosequece of the relatio (4) betwee G ad H. Acowledgmets We are idebted to Mireille Bousquet-M lou for helpful discussios simplifyig the secod part of this wor. : 2

7 Refereces 1. P. Flajolet ad R. Sedgewic. The average case aalysis of algorithms: coutig ad geeratig fuctios. Techical Report 1888, INRIA, D. E. Kuth. The art of computer programmig. Addiso-Wesley Publishig Compay, Readig, MA, Volume 3: Sortig ad searchig. 3. J. W. Moo. Coutig labelled trees. Caadia Mathematical Cogress, Motreal, Que., Caadia Mathematical Moographs, No A. Postiov. Itrasitive trees. J. Combi. Theory Ser. A, 79(2):360366, H. Pr fer. Never Beweis eies Satzes ber Permutatioe. Arch. Math. Phys. Sci., 27:742744, H. S. Wilf. Geeratigfuctioology. Academic Press Ic., Bosto, MA, secod editio, 1994.