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JOURNAL OF COMBINATORIAL THEORY 9, 40 1- 41 1 (19 70 )

Correspondences be tween P lane Trees and B inary Sequences

DAVID A. K L A R N E R *

The University, Reading, England

Communicated by Gian-Carlo Rota

R e c e i v e d J u l y I , 1 9 6 7

A B S T R A C T

T h e s u b j e c t o f e a c h o f t h e f iv e s e c t i o n s o f t h i s p a p e r i s t h e p l a n t e d p l a n e t r e e sd i s cu s s e d b y H a r a r y, P r i n s, a n d Tu t t e [ 7]. A d e s c r i p t i o n o f t h e c o n t e n t o f t h ep r e s e n t w o r k i s g i v e n in S e c t i o n 1 . S e c t i o n 2 is d e v o t e d t o a d e f i n i t i o n o f p l a n et r e e s i n t e r m s o f f in i te se ts a n d r e l a ti o n s d e f in e d o n t h e m - - w e h o p e t h i s d e -f i n i t io n w i l l r e p l a c e t h e t o p o l o g i c a l c o n c e p t s i n t r o d u c e d i n [ 7] . A o n e - t o - o n ec o r r e s p o n d e n c e b e t w e e n t h e c l a s s e s o f i s o m o r p h i c p l a n t e d p l a n e t r e e s w i t hn + 2 v e r t i c e s a n d t h e c l a s s e s o f i s o m o r p h i c 3 - v a l e n t p l a n t e d p l a n e t r e e s w i t h2 n q - 2 v e r t ic e s is g i v e n in S e c t i o n 3 . S e c t i o n s 4 a n d 5 d e a l w i t h e n u m e r a t i o np r o b l e m s .

1 . I N T R O D U C T I O N

H a r a r y , P r in s , a n d T u t t e [7 ] g a v e a c o m p l i c a t e d o n e - t o - o n e c o r r e -s p o n d e n c e b e t w e e n t h e s e t o f c la ss es o f i s o m o r p h i c p l a n t e d p l a n e t r e eswi th n J r 2 ve r ti c e s and t he s e t o f c la s se s o f i som orp h ic t r i va l en t p l an t edp l a n e t re e s w i t h 2 n + 2 v e r ti c es . A l s o , t h e y s h o w e d t h a t t h e n u m b e r o fc la s se s o f i som orp h ic p l a n t ed p l an e t r ee s w i th n q - 2 ve rt i ce s i s(~n)/(n q -1)fo r n = 0 , 1 . . . . S oo n a f t e r t he app ea ran ce o f t h i s pap e r a s imp le r co r r e -

s p o n d e n c e w a s f o u n d [8 ], a n d D e B r u i j n a n d M o r s e l t [3] p r o v i d e d t h r e em o r e s i m p le c o r r e s p o n d e n c e s b e t w e e n t h e se s e ts .

B r i e fl y de sc r ibed t he c on t en t o f t h i s pa pe r i s a s fo l lows : I n Sec t i on 2a new de f in i t i on o f p l an e t r ee s is g iven i n t e rms o f b ina ry r e l a t i ons o nf in it e s e ts ; u s ing t h i s de f in i t i on i t becom es pos s ib l e t o de f ine i som orp h i smo f p l a n e t r e es i n t e r m s o f p e r m u t a t i o n s o f f in i te se ts in s t e a d o f h o m e o -m o r p h i s m s o f t h e p l a n e o n t o i ts el f. T h u s , t h e t o p o l o g i c a l a p p r o a c h o f [7] h a sb e e n a b a n d o n e d h e r e . I n S e c t i o n 3 w e a s s ig n a b i n a r y s e q u e n c e o f l e n g th2n t o each c l as s o f i so m orp h ic p l an t ed p l ane t r ee s w i th n + 3 ve rt i ce s ;

such a s equence (b l . .. .. b2 ,) con t a in s exac t l y n un i t s and i s cha ra c t e r i zedt T h e e d i t o r s r e g r e t t h a t t h e p u b l i c a t i o n o f t h i s p a p e r h a s b e e n d e l a y e d a s a r e s u l t

o f t he ch an ge in ed i to r i a l o ffices .* T h e p a p e r w a s w r i t t e n w h i l e t h e a u t h o r w a s a p o s t d o c t o r a l f e ll o w ( 1 96 7 ) a t M c M a s t e r

U n i v e r s i ty, H a m i l t o n , O n t a r io , C a n a d a .

401

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402 KLARNrR

b y t h e p r o p e r t y t h a t b l + " '" + b 2j > / j f o r j = 1 . ... n . T h e s e b i n a r ys e q u e n c e s c a n a l s o b e a s s ig n e d i n a n a t u r a l w a y t o c la ss e s o f i s o m o r p h i ct r i v a l e n t p l a n t e d l ~]a ne tr e e s w i t h 2 n + 4 v e rt ic e s , a n d i n t h i s w a y a o n e - t o -o n e c o r r e s p o n d e n c e b e t w e e n t h e t w o s et s o f c la s se s o f i s o m o r p h i c t r e e s i sa c h i ev e d . F u r t h e r m o r e , i t i s s h o w n t h a t b i n a r y s e q u e n c e s o f l e n g t hk n c a nb e a s s ig n e d t o t h e c la s se s o f i s o m o r p h i c ( k + 1 ) - v al e n t p l a n t e d p l a n e t r e esw i t h k n + k q - 2 ve r t i ces ; such a sequ enc e (b l . .. .. bkn) con ta ins exa c t ly nu n i t s a n d i s c h a r a c t e r i z e d b y t h e p r o p e r t y t h a t b l + . .. + b k~. ~ j f o rj ~ - 1 .. .. n . I n S e c t i o n 4 w e d e s c r i b e a s i m p l e m e t h o d f o r e n u m e r a t i n gc l as s es o f i s o m o r p h i c p l a n t e d p l a n e t r ee s i n w h i c h t h e d e g r e es o f th e v e r ti c esb e l o n g t o a g i v e n s et o f n a t u r a l n u m b e r s ; f o r e x a m p l e , w e sh o w t h a t t h en u m b e r o f c l as s es o f i s o m o r p h i c ( k + 1 ) -v a le n t p l a n t e d p l a n e t r ee s w i t hk n -k 2 ve r t i ces i s(~n) / (kn - - n + 1 ) . A l s o , w e d e t e r m i n e t h e n u m b e r o fw a y s o f d r a w i n g p l a n e t r ee s s o t h a t t h e e d g e s a n d v e r ti c es a r e m a p p e di n t o t h e e d g e s a n d v e r ti c es o f c e r t a i n n e t w o r k s . W e c o n c l u d e i n S e c t io n 5w i t h s o m e c o m b i n a t o r i a l i d e nt it ie s t h a t a r e a b y - p r o d u c t o f t h e p r o b l e m sc o n s i d e r e d i n t h e e a r l i e r s e c t i o n s .

2. DEFINITIONS AND NOTATION

L e t ( V, E ) d e n o t e a t r e e ; V a n d E d e n o t e t h e s e t o f v e r t ic e s a n d t h e s e to f e d g e s o f t h e t r e e , r e s p e c t i v e l y ; t h e e d g e s o f t h e t r e e a r e 2 - s u b s e t s o f t h ever t i ces . A r o o t e d t r e e ( V, E , v ) i sa t r e e ( V, E ) w i t h a d i s t i n g u i s h e d v e r t e xv e V ca l led the r o o t ; a r o o t e d t r e e i s ap l a n t e d t r e e i f t h e d e g r e e o f t h er o o t i s 1. S u p p o s e ( V, E , v ) is a r o o t e d t r e e a n d l e tp ( x ) d e n o t e t h e l e n g t ho f t h e p a t h f r o m v t o x ~ V; i n p a r t i c u l a r ,p ( v ) = O . A p l a n t e d p l a n e t r e e( V, E , v, R ) i s a p l a n t e d t r e e (V, E , v ) w i t h a l i n e a r o r d e r r e l a t i o n R d e f i n e do n V p o s s e s si n g t w o p r o p e r t i e s:

( i) I f x , y s V an dp ( x ) < p ( y ) , t h e n ( x , y ) ~ R .

(i i) I f {r, s}, {x, y} ~ E w ithp ( r ) = p ( x ) = p ( s ) - - 1 = p ( y )- - 1 , and( r, x ) ~ R , th en ( s , y ) ~ R .

L e t P ( V ) d e n o t e t h e s e t o f p l a n t e d p l a n e t r e es w i t h v e r t e x s et V.T h e d e f in i ti o n o f a p l a n t e d p l a n e t r e e su g g e s ts a c o m p l i c a t e d s t ru c t u r e ;

h o w e v e r, a ll o f t h e i n f o r m a t i o n w e n e e d c a n b e r e c o r d e d i n a n e l e g a n td i a g r a m . T o d r a w a p l a n t e d p l a n e t r e e (V, E , v, R ) w e a r r a n g e t h e v e r t ic e si n r o w s s o t h a t x E V i s in t h e r o w n u m b e r e d p ( x ) ; a l so , th e v e r t ic e s i n e a c hr o w a r e o r d e r e d f r o m l e ft t o r ig h t a c c o r d i n g t o t h e l in e a r o r d e r r e l a t i o n R .A n e d g e {x , y } E E is i n d i c a t e d b y d r a w i n g a s t r a i g h t l i n e f r o m x t o y ; i tt u r n s o u t t h a t a l l o f t h e e d g e s o c c u r b e t w e e n c o n s e c u t i v e l e v el s o f v e r t ic e s ,a n d n o e d g e s c r o ss . A n e x a m p l e o f a d i a g r a m o f a p l a n t e d p l a n e t r e e

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(V, E, a , R ) is g iv en in F igu re 1; in th is t re e V = {a . .. . g} , a i s the r oo t ,E = {{a, b}, {b, c}, {b, d}, {d, e}, {d, f} , {d, g}}, a n d R is d ef in ed b ya < b < " . < g .

F IG . 1 . D i a g r a m o f a p l a n t e d p l a n e tr e e .

C e r t a i n s u b s e t s o fP ( V ) a r e o f sp e c i a l i n t e re s t . L e t D d e n o t e a s u b s e to f th e n a t u r a l n u m b e r s w i t h 1 ~ D , a n d d e fi n eP( V, D)t o b e t h e s u b s e t o fP ( V ) c o n t a i n i n g t r e e s w l ao s e v e r t i c e s h a v e d e g r e e s b e l o n g i n g t o D . I np a r t i c u l a r, w h e n D = {1 , k } t h e e l e m e n t s o fP(V, D ) a r e s a i d t o b ek-valent;o f c o u r se , w h e n D i s t h e s e t o f al l n a t u r a l n u m b e r s w e h a v eP ( V ) = P ( V, D ) .

N o w w e d e f i n e a n e q u i v a l e n c e r e l a t i o n o nP ( V ) c a l l e d i s o m o r p h i s m .Tw o p l a n t e d p l a n e t r e e s ( V, E , v, R ) a n d ( V, F, w, S ) a r eisomorphici fa n d o n l y i f t h e r e e x is ts a p e r m u t a t i o n rr o f V s u c h t h a t

( i ) ~ v = w ,

( ii) {Trx, ry} E F fo r al l {x, y} ~ E , an d

(ii i) (Trx, ~ry) ~ S fo r al l (x, y ) ~ R.

W e l et P * ( V )a n d P *(V , 1:))d e n o t e t h e c la ss e s o f i s o m o r p h i c p l a n t e d p l a n et r e e s d e f i n e d o n t h e s e t sP ( V )a n d P(V, D ),respec t ive ly.

F i n a l l y, i n S e c t i o n 3 w e w i ll n e e d c e r t a i n s e t s o f b i n a r y s e q u e n c e s . L e tB(n, k)d e n o t e t h e s e t o f al l b i n a r y s e q u e n c es(bl ..... b~,,)o f l e n g t h knco nta ini ng ex act ly n un i ts su ch th at b1 + . . . + bk~ ~ j f or j : 1 . .. . n .T h e e l e m e n t s o fB(n, k)c o r r e s p o n d in a n a t u r a l w a y t o v o t in g r e c o r d s inw h i c h o n e c a n d i d a t e h a s a l w a y s a s c o r e a t l e a st k - - 1 t im e s l a rg e r t h a n t h es c o re o f h i s o p p o n e n t . T h i s g e n e r a l i z a t io n o f t h e b a l l o t p r o b l e m w a ss tu d i e d b y D v o r e t z k y a n d M o t z k i n [4]. T h e c a s e k = 2 a p p e a r s i n F e l le r

[ 6 ] , a n d a c o n n e c t i o n w i t h l a t t i c e p a t h s i s e s t a b l i s h e d . F u r t h e r m o r e , F e l l e rg iv e s a s i m p l e p r o o f th a t

2n[ B(n - - 1 , 2)1= ( n ) / ( n + 1). (1)

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T h e C a t a l a n n u m b e r s(~;")/(n + 1 ) a p p e a r i n m a n y c o m b i n a t o r i a l p r o b ,

l e m s ; f o r e x a m p l e , B r o w n [1 ] h a s l i s t e d 4 6 r e f e re n c e s t o p a p e r s i n v o l v i n gthese nu m be rs . Th e ea r l i e s t r e fe ren ce is to Eu le r. M ul l in [12, 13, 14]e n c o u n t e r e d t h e se n u m b e r s i n h is i n v e s t ig a t i o n s o f t r i a n g u l a r m a p s ,w h i c h s u g g e s t s a c o n n e c t i o n b e t w e e n p l a n e t r e e s a n d t h e s e p l a n a rg r a p h s .

3. SOM E ON E-TO -ONE CORRESPONDENCES

I t is e a s y t o p r o v e (b y i n d u c t i o n f o r e x a m p l e ) t h a t t h e n u m b e r o f v e r ti c esi n a ( k + 1 ) - v a le n t p l a n t e d p l a n e tr e e m u s t b e c o n g r u e n t t o 2 m o d u l o k ;th us , i f D = { 1 , k + l } , t he nP ( V, D ) = ;~ u n l es s [ V [ = k n + 2 f o rs o m e n . S u p p o s e Vi s a s e t w i t h [ V [ =k n + 2, a n d D = {1, k + 1}; n o ww e a r e g o i n g t o c o n s t r u c t a o n e - t o - o n e c o r r e s p o n d e n c e ~ b e t w e e n t h ee l e m e n t s o fB ( n - - 1 , k )a n d P * ( V, D ) .

I f T = ( V, E , v, R ) ~ P ( V , D ) ,t h e n T h a s e x a c t l y n v e r t i c e s w i t h d e g r e ek 1. ( A g a i n t h i s is t r i v i a l a n d c a n b e e s t a b l i s h e d b y i n d u c t i o n . ) L e tv~ . .. .. v,~ de no te the ve r t i ces w i th deg ree k + 1 in T, a nd sup po se( v i , v i+ ~ ) ~ R fo r i = 1 . .. . n - - 1 . I fp(v i ) -~ r, t h e n t h e r e a r e k v e r t i c e sva , . . . , v~ i n T s u c h t h a t p(v i l ) - - - - p (v ik ) = r q - 1 ,a n d { v i , vi~} ~ Ef o r j = 1 . .. . k ; f u r t h e r m o r e , w e c a n s u p p o s e ( v ~ j, v i(j+ a)) e R f o r j = 1 . .. .k - - 1 . N o w a b i n a r y s e q u e n c e ( ( T ) = ( b I . .. .. b n k -k ) o f l e n g t hn k - - kc a n b e a s s i g n e d t o T a s f o l l o w s :

l l , i f vii h a s d e g r e e k - k 1 ,b ik - i+ j = 0 , o the rw ise . (2 )

I t c a n b e s h o w n t h a t , i fT, T ' E P ( V , D )a r e i s o m o r p h i c , t h e n ~ ( T ) = ~ ( T ' ),s o , f o r a c la s s X o f i s o m o r p h i c ( k + 1 ) - v a l e n t p l a n t e d p l a n e t r e e s , w e c a nd e f in e ~ ( X ) = ~ ( T ) f o r a n y T ~ X . A l s o , i t i s e a s y t o v e r i f y t h a t ~ ( T ) =( b l . .. .. b , ~ _ ~ ) ~ B ( n - 1 , k )s in c e e a c h v e r t e x o f d e g re e k - ? 1 i n Tc o r r e s p o n d s t o a u n i t in th e s e q u e n c e ( h e n c e , ~ ( T ) h a s e x a c t l y n u n i t s) , a n dthe pa r t i a l seque nce (b l , .. ., bjT~) desc r ib es a (k l ) -v a len t p la n te dp l a n e t r e e h a v i n g a t l e a s t j v e r t i c e s o f d e g r e e k 1 i n T ( h e n c e ,b l " " + b~-k >~ j ) . O b v i o u s l y, n o n - i s o m o r p h i c t r e e s a r e a ss i g n e d d if f e re n tb i n a r y s e q u e n c e s b y ~ . F i n a l l y, t h e r e i s a s i m p l e , o b v i o u s c o n s t r u c t i o nw h i c h s h o w s t h a t a g i v e n s e q u e n c e(b l , .. . , b ,~k-k ) ~ B (n - - 1 , k )c o r r e s p o n d St o s o m e t r e e inP ( V, D ) , s o ~ i s a o n e - t o - o n e c o r r e s p o n d e n c e .

L e t n d e n o t e a n a t u r a l n u m b e r , a n d l e t W d e n o t e a s et w i t h I W I =n q - 2 . N o w w e c o n s t r u c t a o n e - t o - o n e c o r r e s p o n d e n c e X b e t w e e nP * ( W )

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CORRESPONDENCES BETWEEN PLANE TREES AND BINARY SEQUENCES 4 0 5

a n d B ( n - - 1, 2 ). S u p p o s e T e P ( W ) w i t h T = ( W , E , w , R ) , le tW = {w, wt . .. .. wn+x} , an d su pp o se R is de f ine d by w < w 1 < . .. < w n+ l .L e t d~ d e n o t e t h e d e g r e e o fw i f o r i = 1 ,.. ., n 4 - 1, a n d n o t e t h a t d l ~ 2w h i l e d ,~ + l = 1 . W e a s s o c i a t e a b i n a r y s e q u e n c e ( b i l , .. ., b ia ,) o f l e n g t h d iw i t h w~ f o r i ~ - 1 .... n 4 - 1 w h e r eb it = 1 f o r j- -- -- 1 , .. ., d i - - 1 , an dbia, = 0 . A b i n a r y s e q u e n c e ~ ( T ) = ( b l , .., b 2~ _~ ) c a n n o w b e d e f i n e d b yp u t t i n g (bo , bl , . . . , b~n) ~ (b ~ . . .. , b~a1 .. . . . b(~+~)~ .. . . .b (, +~ ) d,+ ) . N o t e t h a tbo ~- 1, b2~-1 = b~a ,- = 0 , a n d bzn ~ b(n+l)a,+~ -~ 0f o r a l l T ~ P ( W ) , s ot h e se b i ts d o n o t a p p e a r i n ~ ( T ) . I t is e a s y t o s e e t h a t t h e l e n g t h o f t h es e q u e n c e ~ ( T ) i s i n d e e d 2 n - - 2 s i n c e d t 4 - "-. 4 - d n + l - - 3 = 2 ( n - - 1 ).A l s o , t h e z e r o s i n ~ ( T ) c o r r e s p o n d o n e - t o - o n e t o t h e v e r t ic e s w I .... . w,~ _ l,s o ~ ( T ) h a s e x a c tl y n - - 1 u n i t s w h i c h c o r r e s p o n d o n e - t o - o n e t o t h e e d g e si n t h e s e t E ' ~ - E l { { w , w l }, { w a ,w~}} : F in a l ly , n o te th a t b l -4- " '" 4 - b~. > / jb e c a u s e ( b a . . . . . b 2 j ) d e s c r i b e s a p o r t i o n o f T i n v o l v i n g a t l e a s t j e d g e s o f Tb e l o n g i n g t o E ' . T h u s , ;~ is a m a p p i n g o fP ( W ) i n t o B ( n - - 1 , 2 ) ; f u r t h e r -m o r e , i t is o b v i o u s f r o m o u r c o n s t r u c t i o n t h a t ~ is o n t o . I fT, T ' e P ( W ) ,t h e n w e h a v e ~ (T ) =x ( T ' ) i f a n d o n l y if T a n d T ' a r e is o m o r p h i c . T h u s ,f o r e v e r y X e P * ( W ) w e c a n d e f i n e x ( X ) = y ~ ( T ) f o ra n y T ~ X , a n d X isa o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e nP * ( W ) a n d B ( n -- 1 , 2) .

L e t n d e n o t e a n a t u r a l n u m b e r , a n d l et V, W d e n o t e s e ts w i t h I V ] ----2 n 4 - 2 , I W I ---- n 2 . W e h a v e a o n e - t o - o n e c o r r e s p o n d e n c e X b e t w e e nP * ( W ) a n d B ( n - - 1, 2 ); a l so , w e h a v e a o n e - t o - o n e c o r r e s p o n d e n c eb e t w e e n P * ( V, {1, 3}) and B ( n - - 1 , 2 ). C o m b i n i n g ~ a n d X i n t h e u s u a lw a y w e o b t a i n a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e nP * ( V, {1, 3}) andP * ( W ) . I n F i g u r e 2 w e h a v e d r a w n d i a g r a m s r e p r e s e n t i n g t h e c l as s es o fi s o m o r p h i c t r iv a l e n t p l a n t e d p l a n e t re e s w i t h e i g h t v er ti ce s , a n d t h e c la s se so f is o m o r p h i c p l a n t e d p l a n e t re e s w i t h f iv e v e r ti c e s; t h e b i n a r y s e q u e n c e sa s s i g n e d t o t h e s e t r ee s b y ~ a n d X h a v e a l so b e e n i n d i c a te d .

( 1 , 0 , 1 , 0 ) ( 1 , 0 , 0 . 1 ) ( 0 , 1 , 1 , 0 ) ( 0 , 1 , 0 , 1 ) ( 1 , 1 , 0 , 0 )

FIG. 2. Correspondence between trivalent an d ord inary planted plane trees.

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4. ENUMERATION

F o r t h e m o m e n t w e le t D d e n o t e a fi x ed s u b s e t o f t h e n a t u r a l n u m b e r sw i t h l e D ; a ls o , l et V d e n o t e a s et w i t h I v I = n q - 1 a n d d ef in et (n , D) = I P * ( V ,D) [. S ince D i s fi xed we can abb rev i a t et (n , D) to t(n) ,a n d l e t T ( x ) ~ t (1 )x + t (2 )x2 + . . . . Fo r a g ivend s D , d v6 1, letP a * ( V, D ) d e n o t e t h e s e t o f e l e m e n t s o fP * ( V, D ) invo lv ing t r ees such t ha tt he deg ree o f t he ve r t ex j o ine d t o t he roo t is d. I t i s c lea r t ha t

# - * ( V, D )[ ----- ~ t ( n l ) ' " t ( n a - 1 ) , (3)

wh e re t he sum ex t ends ove r a ll com pos i t i ons (na , .. ., n~ - l) o f n - - 1 i n toexac t l y d - - 1 pos i t i ve pa r ts . Th us , IP a * ( V, D)[ i s the coeff ic ien t o f x ~ inthe power s e r i e sxTa -~ (x ) , s o w e h a v e

T ( x ) = x + x Y T ~ - ~ ( x ) . (4 )a~ D

I f D = {1, 2 ,. .. }, the nP * ( V, D ) = P * ( V ) ,and (4 ) imp l i e s

T(x ) = x 4 - xT(x ) / (1 - -T(x) ) ; (5 )

t ha t i s ,

T2 (x ) - - T (x )+ x = 0. (6)

Solv ing for T in (6 ) an d us ing the f ac t th a t t (1 ), t (2 ) .... a re pos i t ive we have

1 2 - - 21 ( 1 - - ( 1 - - 4 x ) 1 /2 ) = ~ ~ ( : 1 )X n ' (7 )( x ) =

s o t h e n u m b e r o f c l as se s o f n o n - i s o m o r p h i c p l a n t e d p l a n e t re e s w i t h n q - 2ver t ices i s(~")/(n 4- 1).

I f D = {1, k 4- 1}, th enP * ( V, D ) d e n o t e s t h e s e t o f c la ss es o f is o m o r p h i c(k 4 - 1 ) -va l en t p l a n t ed p l ane t r ee s ; i n t h i s ca se (4 ) becom es

T(x) =- x + xTk (x) . (8)

M a k i n g t h e s u b s ti tu t i o nT(x ) = xU (xk ) , x k = yin (8 ), we ob t a in a r e l a t i one q u i v a l e n t t o

y U k ( y ) - - U ( y )+ 1 = 0. (9)

I t i s kn ow n (see P61ya and Szeg6 [15, p . 125] th a t

(10)

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s o it f o l l o w s t h a t t h e n u m b e r o f c la s se s o f is o m o r p h i c ( k + 1 ) -v a le n tp l a n t e d p l a n e t r e e s w i t hk n + 2 v e r t i c e s i s

= - .

L e t k d e n o t e a n a t u r a l n u m b e r w i t h k > I , a n d s u p p o s e D = { I . .. . . k } ,t h e e l e m e n t s o fP * ( V, D )i n v o l v e p l a n t e d p l a n e t re e s i n w h i c h e a c h v e r t e xh a s d e g r e e 1 , . . . , k ; i n t h i s c a s e ( 4 ) b e c o m e s

T ( x ) = x + x ( T ( x ) -+- " " +T k - l ( x ) ) , ( 11 )

w h i c h i m p l i e s

T ( x ) ( 1 - - T ( x ) ) = x ( 1 - - T k ( x ) ) . (12 )

W e h a v e b e e n u n a b l e t o d e t e r m i n e a p l e a s a n t f o r m u l a f o r t h e c o e ff ic ie n t o fx '~ i n t h e p o w e r s e r ie s w h i c h s a ti s fi e s ( 1 2 ). F o r e x a m p l e , c o n s i d e r t h e c a s ek = 3 , t h e n ( 11) b e c o m e s

x r ~ ( x ) - ( 1 - x ) r ( x ) + x = 0 , ( 1 3 )

a n d t h i s i m p l i e s

T ( x ) = (1 - - x - - (1 - - 3x ) l / z (1 +x ) l / ~ ) / a x . (14 )

U s i n g ( 1 4 ) , a n e x p l i c i t f o r m u l a f o rt ( n ) , t h e c o e f f i ci e n t o f x " i n th e p o w e rs e ri e s r e p r e s e n t a t i o n o fT ( x ) , c a n b e f o u n d , b u t t h is e x p r e s s i o n d o e s n o ta p p e a r t o h a v e a s i m p l e f o r m . U s i n g (1 3 ), w e c a n f i nd a r e c u r r e n c e r e l a t i o ns a t i s f ie d b y t h e s e q u e n c e { t ( n ) : n = 1 , 2 , . . } ; t h i s i s d o n e b y d i f f e r e n t i a t i n gt h r o u g h ( 1 3 ) a n d t h e n r e p l a c i n gx T ~ ( x ) w i t h ( 1 - x ) T ( x ) - x ; t h er e s u l t i n g r e l a t i o n im p l i e s t ( 0 ) = 0 , t (1 ) = 1 , a n d

t ( n ) = { (2n - - 1 )t ( n - - 1) - ? 3 (n - - 2 )t ( n - - 2)}/ (n + 1) (15)

f o r n = 2 , 3 . . . . . T h u s ,t ( n ) is 1, 1, 2, 4, 9 . . . . fo r n = 1, 2, 3, 4, 5 . . .. re sp e c-t iv e ly . D i a g r a m s r e p r e s e n t i n g t h e n i n e cl a ss e s o f i s o m o r p h i c p l a n t e d p l a n et r e e s w i t h s i x v e r t i c e s s u c h t h a t t h e d e g r e e o f e a c h v e r t e x i s 1, 2 , o r 3 a r eg i v e n i n F i g u r e 3 .

FI6. 3. Planted pla ne trees in wh ich vert ices hav e degree 1, 2, or 3.

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N o w w e c o n s id e r a p r o b l e m c o n c e r n i n g th e n u m b e r o f e m b e d d i n g s o f

c e r t a in p l a n t e d p l a n e t re e s i n n e t w o r k s . W e b e g i n w i t h a n e x a m p l e . L e tI de no te the se t o f G aus s ian in tegers , l e t U = {{x , y} : x , y E I , ] x - y ] = 1},th en C = ( I , U , {0, 1}) i s an exa m ple of anedge-rooted linear graph; Iisthe ve r tex se t, U is the edge se t , and {0, 1} ~ U i s th e ro o t ed edge o f C . Le tT = (V, E , v, R) de no te a p lan ted p lan e t ree , th en , i f {v, w} ~ E , (V, E ,{v, w }) is a n o t h e r e x a m p l e o f a n e d g e - r o o t e d l i n e a r g r a p h . A m a p p i n g50 sen din g V in to I is anembeddingof T i n C i f

(i) 50v = 0, 50w = 1,

(ii) {a, b} E E im pl ies {50a, 50b} ~ U, a n d

(iii) {a, b}, {a, e} ~ E w ith b =~ e im pl ies 50b @ 50c.

N o t e t h a t an em bed d ing 50 i nduces a m app ing cp' s end ing edges o f T i n tothe edges o f C ; a l so , 50' s ends ad j a cen t edges o f T t o ad j acen t edges in C ;f i nal l y, 50' i s no t neces sa r i l y one - to -one . T hus , T can be em bed ded i n Ci f and on ly i f eve ry ve r t ex i n T has deg ree 1 , 2 , 3 , o r 4. Em bedd ings o fp l a n t e d p l a n e t r e e s s u ch t h a t e a c h v e r t e x o f T h a s d e g r e e 1, 2 , 3 , o r 4 a ri s ena tu ra l l y i n Ed en ' s [5 ] i nves t i ga t i ons o f t he ce ll g row th p rob l em . (A ne x p o s i t i o n o f E d e n ' s w o r k a l s o a p p e a r s i n [9 ].) Tw o - d i m e n s i o n a l l a tt ic ew a l k s c a n b e v i e w e d a s e m b e d d i n g s o f th e p l a n t e d p l a n e t r e e i n w h i c heve ry ve r t ex ha s deg ree 1 o r 2 .

Su pp ose V i s a se t w i th I V] = n q- 1 , l e t D = {1, 2 , 3 , 4}, an d cons id erthe s e t o f t re e sP(V, D) . W e l e t A (T ) d e n o t e th e n u m b e r o f em b e d d i n g s o fT e P ( V, D )in C . I t i s c lea r tha t , i fT, T ' e P ( V, D )a r e i s o m o r p h i c , t h e nA(T) = ;~(T') . T hu s, fo r a give nX e P * ( V, O ),we can de f ineA(X) --= A(T)fo r a ny T E X . F ina l l y, w e de f ine

a(n) = ~ A( X) , (16)XeP*(V,D)

a n d l e t A(x) = a(1) x q- a (2)x z + "" . Clear ly,

A(x) = x + 3xA(x) + 3xA~(x) + xA3(x), (17)

so , a f te r se t t ingB(x) = 1 + A(x) ,(17 ) becomes

xB~(x) -- B(x) + 1 = O.

Using ( i 0 ) , we s ee t ha t ( 18 ) imp l i e s

( 1 8 )

1 3nA(x) = ~ n ( n - 1) xn ' (19)

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S O

3n (2na ( n ) = ( n ) / + 1).

Inc iden t a l l y, th i s shows tha ta (n ) is a ls o t h e n u m b e r o f c la ss e s o f i s o m o r p h i cp l a n t e d p l a n e t re e s w i t h 3 n + 2 v e r ti c es . D i a g r a m s r e p r e s e n t in g t h ee m b e d d i n g s o f t h e c la s se s o f i s o m o r p h i c p l a n t e d p l a n e t r e e s w i t h 4 v e r ti c esa r e s h o w n i n F i g u r e 4 .

El I 0 I 0 I 0 1

O I 0 1 0 I

0 I 0 I 0 I

FIG. 4.

0 1

Em beddings o f planted plane trees in the sq uare lattice.

I f w e h a v e D = { 1 , . . . , k + 1}, a n e l em e n tT ~ P ( V, D ) c a n b e" e m b e d d e d " i n t h e p l a n e s o t h a t t h e e d g e s o f T a r e a l l o n e u n i t l o n g a n dp a r a ll e l t o o n e o f t h e l in e s w h i c h m a k e s a n a n g l e o f2r r j / (k + 1) wi th thex - ax i s f o r j -~ 0 ,. .. , k . T h e n u m b e r o f e m b e d d i n g s o fP * ( V, D )in the senseo f ( 1 6 ) tu r n s o u t t o b e(~ '~) / (kn - - n +1) - - t h i s i s al so t he nu m be r o f c la s se so f i s o m o r p h i c ( k + 1 ) -v a le n t p l a n t e d p l a n e t re e s w i t hk n + 2 ver t ices .

5. T w o COMBINATORIAL IDENTITIES

I t w a s s h o w n i n t h e l as t s e c t io n t h a t t h e n u m b e r o f c la ss es o fi som orph ic (k + 1 ) -va len t p l an t ed p l an e t r ee s w i thk n + 2 ver t ices is( ~ ) / ( k n - - n +1); how eve r, t h i s s e t o f tr e e s can be enu m era t ed i n a s econd

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410 KLARNER

w a y t o o b t a i n t h e i d e n ti t y

k n - - n = ~ "'"n ~ / \ n 3 / ~ n j ] '( 2 0 )

w h e r e t h e s u m e x t e n d s o v e r a ll c o m p o s i t io n s( / / 1 . . . . ,n j ) o f n in t o a n u n r e s -t r i c t e d n u m b e r o f p o s i t i v e p a r t s w i t h n~ ~-- 1. W e p r o v e ( 2 0 ) a s f o l l o w s :L e t S(n ~ . . .. , n j )d e n o t e t h e s et o f c l a ss es o f i s o m o r p h i c ( k + 1 ) - v a le n tp l a n t e d p l a n e t r ee s in w h i c h t h e n u m b e r o f v e r ti c es x s u c h t h a tp ( x ) = i + 1is k / / i f o r i ~ - 1 ... . j . I n d r a w i n g r e p r e s e n t a t i v e e l e m e n t s o fS(n~ . . . . . / /5 )w e n o t e t h a t t h ek n i v e r t ic e s in t h e i - th l ev e l c a n b e j o i n e d t o t h ek n i + lver t ices in the ( i + 1)-s t level in ex ac t ly (kn~ ~ w ay s f o r i = 1 . .. . j - - 1 .\ni+l~'T h u s ,

I s (n . . . . . n , ) l = . . . ( 2 1 )\ n 2 / \ n a / I n j } '

a n d s u m m i n g o v e r a p p r o p r i a t e c o m p o s i t i o n s o f n g iv e s (2 0).I t w a s a l s o s h o w n i n S e c t i o n 4 t h a t t h e n u m b e r o f c l as s es o f i s o m o r p h i c

p l a n t e d p l a n e t r e e s w i t h n + 2 v e r t i c e s i s(~n ) / (n + 1) , bu t th i s se t can

a l s o b e e n u m e r a t e d i n a s e c o n d w a y t o o b t a i n t h e i d e n t i t y

1 + n 2 - 1 ) ( n 2 - + - n 3 - 1 ) (F /j_- ~ - n j - 1 ) , ( 2 2 )( n + 1 ) ( 2 2 ) = } -~ '(H1 1 / 2 / / 3 " " / / J

w h e r e t h e s u m e x t e n d s o v e r a ll c o m p o s i t i o n s( n I . . . . . n~) o f n 1 in to anu n r e s t r ic t e d n u m b e r o f p o s i ti v e p a r t s w i t h n l = 1. To p r o v e (2 2 ) l e tP(n l . . . . . n~ )d e n o t e t h e n u m b e r o f c la ss es o f i s o m o r p h i c p l a n t e d p l a n et r e e s i n w h i c h t h e r e a r e e x a c t l y n i v e r t i c e s x s u c h t h a tp ( x ) = i f o r

i = 1 , . .. ,j . To d r a w t h e r e p r e s e n t a t i v e e le m e n t s o fP ( n i . . . . . n j )n o t e t h a tt h e n i v e r t ic e s i n t h e i - th l ev e l c a n b e j o i n e d t o t h e n i+ l v e r t ic e s in t h e( i + 1 ) - st l eve l in jus t

n i - ] - h i+ 1- - 1 )/ / i + i

w a y s . T h u s ,

~ ' " ~ 1 72 /7 3 / / j( 2 3 )

a n d s u m m i n g ( 23 ) o v e r t h e a p p r o p r i a t e c o m p o s i t i o n s o f n + 1 y i el d s ( 22 ).A l l o f t h e r e su l ts p r e s e n t e d i n th i s p a p e r f o r m e d a p a r t o f t h e a u t h o r ' s

t h e s is [8 ]; a l so , a t h e o r y f o r s u m s h a v i n g t h e f o r m o f (2 0 ) o r (2 2 ) a p p e a r s

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i n [ 1 0] . R e c e n t l y, t h e p a p e r o f C a r l i tz [ 2 ] w h i c h d e a l s w i t h R i o r d a n ' s [ 16 ]

r e s u l t s o n c h r o m a t i c t r e e s s t i m u l a t e d m e t o u s e t h e m e t h o d s o f S e c t i o n 3t o o b t a i n a o n e - t o - o n e c o r r e s p o n d e n c e b e t w e e n c e r ta in c h r o m a t i c tr e esa n d ( k q - 1 ) - v a l e n t p l a n e tr e e s ; s e e [ 11 ].

R E F E R E N C E S

1 . W . G . B R OW N , H i s t o r ic a l N o t e o n a R e c u r r e n t C o m b i n a t o r i a l P r o b l e m ,A m e r.M a t h . M o n t h l y7 2 ( 1 9 6 5 ) , 9 7 3 - 9 7 7 .

2 . L . C A R LIT Z, A N o t e o n t h e E n u m e r a t i o n o f L i n e C h r o m a t i c Tr ee s ,J . Com binator ia lTheory 6 (1969) , 99 -101 .

3 . N . G . DE BRUIJN AND B. J . M . MORSELT, A N ote on P la ne Trees ,J . Combinator ia lTheory 2 (1967) , 27 -34 .

4 . A . D V OR ET ZK Y A ND T. M O T ZK IN , A P r o b l e m o f A r r a n g e m e n t s ,D u k e M a t h . J .14 (1947) , 305-313 .

5 . M . E D EN , A T w o - D i m e n s i o n a l G r o w t h P r o c e ss ,Proceedings of the Fourth BerkeleySympos ium on Ma themat ica l S ta t i st i c s and Probabi l ity, ( J .N e y m a n , e d . ) , Vo l . I V,U n i v e r s i t y o f C a l i f o r n i a P r e s s , B e r k e l e y, 1 96 1, p p . 2 2 3 - 2 3 9 .

6 . W. FELLER,A n Introduction to Pro babili ty Theo ry and l ts Applications,2 n d e d . ,Vo l . 1 , W i l ey, N e w Yo r k , 1 9 6 5.

7 . F. HARARY, G . PRINS, AND W . TUTTE, Th e N u m be r o f P l an e Trees ,Nederl . Akad.Wetensch. Proc. Ser A67 [Indag. Math. 26 (1964) , 319-329] .

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