THE NUMBERS OF LABELED COLORED AND CHROMATIC TREES

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THE NUMBERS OF LABELED COLORED AND CHROMATIC TREES

BY J O H N RIORDAN

in New York N. Y .

1. Introduction

The mathematical theory of trees, as first discussed by Cayley in 1857 [3], was concerned in the enumeration aspect with two contrasting cases; the points (or lines) of the trees in question were either all alike or all unlike. For the allied subject of series-parallel electrical networks, R. M. Foster [5] has introduced enumerations b y two variables, the number of elements in the network (corresponding to lines of a tree) and the number of these which are marked, with each mark distinct from every other. Two networks which differ only by a permutation of marks are counted as different if the differently marked elements are dissimilar, with dissimilarity as ex- plained in [5] and [2]. The same kind of enumeration is done here for trees, thus in- cluding both classical cases in one frame. The trees so marked are called labeled trees.

A second kind of marking is also considered. This is familiar from graph coloring, where each point or line of a graph is given one of c colors. Note t h a t every element (point or line) of a colored tree has some color (if uncolored elements were permitted, they could all be said to be colored with a new color c + 1) and t h a t in a n y particular coloring, a n y number of different colors (up to c) m a y appear, in con- trast to a labeled tree. The enumeration is by number of elements (points or lines) and b y number of colors, or b y number of elements and b y number of distinct colors.

I t m a y be noted t h a t the enumerating functions for colored trees are formally similar to those for unmarked trees, while both differ from those for labeled trees.

This formal similarity does not persist when tree colorings are subject to the chromatic condition t h a t adjacent elements (points having a line in common or lines having

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212 J O H N R I O R D A N

a point in common) m a y not have the same color, the third kind of marking considered here. Colored trees satisfying the chromatic condition are here called chromatic.

For each of these three kinds of murking, the trees considered m a y be rooted or not, oriented or not, and the marked elements m a y be points or lines; hence here are 24 possible enumerations, all of which are carried out here.

This large a m o u n t of material is given a unified exposition by means of a theorem due to Pdlya (the Hauptsatz of [8]) for rooted trees, and a combination of the work of Pdlya (1.e.) and t h a t of Otter [7] for the free (unrooted) trees. I t m a y be noticed t h a t the labeled eases do not permit the use of Pdlya's theorem in its stated form, b u t the proof given by Pdlya is in fact sufficiently general to permit its application also to these eases; this is also the ease for the labeled series-parallel networks treated in [2]. I t m a y be noted also t h a t the same procedure flows with equal directness for the various kinds of linear graphs in H a r a r y [5] when these are marked in a n y of the three ways, but for brevity these results are deferred to a later occasion.

Generating functions are used here, as in other combinatorial settings, for convenience in expressing briefly the relationships of their coefficients, which here are numbers of the various kinds of trees in question. These relationships in principle could have been derived in a strictly finite combinatorial way. I n this usage, the variables of the generating functions are to be regarded as indeterminates or tags whose powers identify the coefficients, and formal operations on the generating functions used to express coefficient relations are replacements of basic rules in an under- lying algebra of sequences, as in Bell [1].

2. Definitions

The definitions necessary for clarity as to what is being enumerated are given in this section. Because they are mostly well known, they are given with a minimum of explanation.

A tree is a connected linear graph without cycles (or slings), hence of nullity (Zusammenhangszahh Pdlya) zero. Because of the absence of cycles, the number of points in a tree is one greater than the number of lines, and either m a y be used as a parameter. A rooted tree is a tree in which one point, the root, is distinct from the others; when the root is connected to only one other point, the tree is called planted (Setzbaum: Pdlya). An oriented tree is a tree in which every line is directed.

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T H E N U M B E R S O F L A B E L E D C O L O R E D A N D C H R O M A T I C T R E E S 213 Two p o i n t s of a t r e e are a d j a c e n t if t h e y h a v e a line in c o m m o n , t w o lines if t h e y h a v e a p o i n t in c o m m o n .

A b r a n c h of a t r e e a t a n y p o i n t consists of a line t o a n a d j a c e n t p o i n t a n d all t h e p o i n t s a n d lines w h i c h m a y be r e a c h e d b y p a t h s f r o m t h e g i v e n p o i n t t h r o u g h t h i s line; t h e b r a n c h e s a t a r o o t are all p l a n t e d trees. T h e

weight

of a b r a n c h is t h e n u m b e r of lines i t c o n t a i n s ; t h e w e i g h t of a p o i n t is t h e w e i g h t of its l a r g e s t b r a n c h . T h e c e n t r o i d ( M a s s e n z e n t r u m ) of a t r e e is t h e set of all p o i n t s of s m a l l e s t weight;

i t is well k n o w n t h a t t h i s set consists e i t h e r of a single p o i n t , w h e n t h e t r e e is s a i d t o b e ( u n i ) c e n t r o i d a l , or of t w o a d j a c e n t p o i n t s , w h e n i t is s a i d t o be b i c e n t r o i d a ] . I t is also k n o w n t h a t b i c e n t r o i d a l trees w i t h a n o d d n u m b e r of p o i n t s a r e i m p o s s i b l e , while for a n e v e n n u m b e r of p o i n t s , p - 2 q, t h e b r a n c h e s a t each c e n t r o i d , e x c l u d i n g t h e one t h r o u g h t h e o t h e r , h a v e in t o t a l q points. T h e s e f a c t s will b e u s e d in re- ] s t i n g t h e e n u m e r a t i o n of t r e e s t o t h a t of r o o t e d trees.

Two trees are

isomorphic

if t h e r e is a one t o one c o r r e s p o n d e n c e b e t w e e n t h e i r p o i n t s w h i c h p r e s e r v e s a d j a c e n c y . Two r o o t e d trees are i s o m o r p h i c if t h e y a r e iso- m o r p h i c as trees a n d t h e c o r r e s p o n d e n c e l e a v e s t h e r o o t u n c h a n g e d . P o i n t s or lines c a r r i e d i n t o e a c h o t h e r b y a n i s o m o r p h i s m are called similar. Two l a b e l e d trees are i s o m o r p h i c if t h e y a r e i s o m o r p h i c as u n l a b e l e d trees a n d t h e e l e m e n t s l a b e l e d ( p o i n t s or lines) e i t h e r r e m a i n u n c h a n g e d or are c h a n g e d t o s i m i l a r e l e m e n t s . S i m i l a r de- f i n i t i o n s h o l d for c o l o r e d a n d c h r o m a t i c t r e e s as well as for t h e c o r r e s p o n d i n g ori- e n t e d v a r i e t e s .

As in t h e classical case of C a y l e y , all e n u m e r a t i o n s a r e of n o n i s o m o r p h i c t r e e s (of t h e v a r i o u s k i n d s c i t e d a b o v e ) .

N o t e t h a t b e c a u s e l a b e l i n g is d o n e w i t h d i s t i n c t m a r k s , e v e r y i s o m o r p h i s m of a l a b e l e d r o o t e d t r e e carries e a c h l a b e l e d b r a n c h a t t h e r o o t e i t h e r i n t o itself or i n t o one of i t s i s o m o r p h i c images, a n d n e v e r i n t o a n o t h e r l a b e l e d b r a n c h . This is t o s a y t h a t two b r a n c h e s of t h e s a m e l a b e l e d t r e e can be i s o m o r p h i c o n l y w h e n e a c h h.~s no labels.

3. P61ya's Theorem

F o r b r e v i t y of s t a t e m e n t , t h e t h e o r e m will b e s t a t e d in a l i m i t e d t w o v a r i a b l e f o r m w h i c h satisfies p r e s e n t needs. Also P d l y a ' s g e o m e t r i c a l t e r m i n o l o g y will be a b a n - d o n e d . A few p r e l i m i n a r y r e m a r k s are n e c e s s a r y .

T h e t h e o r e m concerns t h e r e l a t i o n s of t w o e n u m e r a t i n g g e n e r a t i n g f u n c t i o n s , for b r e v i t y h e r e c a l l e d e n u m e r a t o r s .

T h e first of t h e s e , S ( x , y), is t h e e n u m e r a t o r of a. store of o b j e c t s a c c o r d i n g to t h e i r r a n k or size w i t h r e s p e c t t o t w o g i v e n c h a r a c t e r i s t i c s e.g. t h e l a b e l e d trees

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2 1 4 J O H N RIORDAN

enumerated b y n u m b e r of points and b y n u m b e r of labels. This enumerator in general is of the form S (x, y) = • Sij/~ (x) gj (y)

with the sets of functions (/4 (x)), (gj(y))linearly independent. The form of these functions is dictated b y the given characteristics. The proper choice for labeled trees, as will appear, is /n ( x ) = x n, gm (y)=y'~/m!.

The second enumerator, T (x, y), is the enumerator, with respect to the same characteristics as t h a t of the store, of the inequivalent selections of the objects n at a time and in order, with each object chosen independently. Thus T (x, y) is of

the form T (x, y) = Z Tij/~ (x) gj (y).

This leaves undefined equivalence of selections and composition of characteristics.

The rules for these are as follows. The first refers to the order of selection and m u s t be preassigned. Two selections are equivalent if there is a p e r m u t a t i o n of a group G which sends one into the other. The group G is specified b y its cycle index

1 ttltl....t

H ( t l , t 2 . . . t n ) = ~ Z h t , i . . . . i n ,

with i i + 2 i 2 + . . . + n i n = n , h the order of G and h~,i .... tn the n u m b e r of p e r m u t a - tions of G having i I cycles of length one, i 2 of length two and so on. Note t h a t Hn (1, 1 . . . 1) = 1, and t h a t if all orders of selection are distinct Hn =t~ ~ while if all are equivalent Hn is the cycle index of the symmetric group, which I write Cn (t 1, t 2 .. . . . tn)/n! and take

1 + ~ Cn (t 1, t 2 .. . . . tn)/n! = exp (t 1 § t2//2 + . . . + tn/n § (1) 1

which can be easily verified from the explicit expression cf the Cn (P61ya [8], p. 162).

The second, the composition of characteristics, applies in the case of a fixed order of selection, and is t a k e n as defined b y the product rule, as in Ford and Uhlen- beck [4]; namely, the enumerator for two objects together with b o t h orders of selection distinct and hence with cycle index H2=t~, say T2(x, y), m u s t satisfy T 2 (x, y ) = S 2 (x, y). This rule and the nature of the characteristics of the objects enu- m e r a t e d determines the functions /~ (x) and gj (y) in the enumerator, S (x, y).

The theorem m a y n o w ' b e s t a t e d as follows:

T H E O R E M (PSlya): I / objects are chosen independently /rom a store o/ ob]ects having enumerator S = S (x, y) which satis/ies the product rule, and i/ the order equi.

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T H E I C U I Y f B E R S O F L A B E L E D C O L O R E D A N D C H R O M A T I C T R E E S 215 valence o/ choices o/ n is speci/ied by the cycle index H~ (tl, t 2 . . . t n ) , then the distinct (inequivalent) choices o/ n have enumerator T~ (x, y ) = Hn (S 1, S 2 . . . S,.) with S~ the enu- merator /or choices o/ Ic identical objects ( ~ - - 1 , 2 . . . S 1 = S).

N o t e t h a t t h e p r o d u c t rule itself becomes a special case of t h e theorem, w h e n H~ = t~.

This differs f r o m P61ya, firstly in t h a t t h e p r o d u c t rule is t a k e n as one of t h e assumptions o f the t h e o r e m which makes it possible to use the more general kind of e n u m e r a t o r s which a p p l y to t h e labeled cases, secondly in t h a t t h e conclusion is s t a t e d '% stage earlier". T h e choices of k i n v a r i a n t for a cycle of length k are exclusively of like objects a n d if

S (x, y) = Z S~s x i yJ,

t h e n Sk = S (x k, yk), which is P S l y a ' s conclusion. P61ya's proof applies to this more general case with slight changes of f o r m u l a t i o n only. As n o t e d above, a different f o r m of e n u m e r a t o r is required for labeled trees.

4. R o o t e d Trees with P o i n t Labels

T a k e rp, m as t h e n u m b e r of (nonisomorphic) r o o t e d trees with p points, m of which are labeled with distinct labels, in t h e w a y first described above. Because t h e labels are distinct, t h e e n u m e r a t o r which satisfies t h e p r o d u c t rule m u s t be t a k e n in t h e f o r m

r (x, y) = ~ rp, ~n xPP Ym/m! 1

= ~ x" Z r,, ,~ y'~/m! = Y. x ~ r~ (y)

p = l rn=0

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T a k e r~,m(n) as the root, so t h a t

p - 1

r~, ,n = ~ rp. m (n). (3)

n = l

N o w a p p l y P S l y a ' s t h e o r e m to t h e e n u m e r a t i o n of rp, m (n) or w h a t in t h e same thing, t o t h e d e t e r m i n a t i o n of t h e e n u m e r a t o r :

rn (x, y) = • r~, ,n (n) x v y m / m !

since t h e assignment of labels to two trees t o g e t h e r involves a binomial coefficient which is p r o p e r l y a c c o u n t e d for b y f o r m (2) in a m a n n e r familiar f r o m t h e generating f u n c t i o n for p e r m u t a t i o n s of objects of general specification.

corresponding n u m b e r w h e n there are n branches at t h e

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2 1 6 J O H N RIORDAN

T h e stpre is t h e collection of p l a n t e d trees which m a y be branches at t h e root, hence has e n u m e r a t o r r (x, y). I n t h e f o r m (2), r (x, y) satisfies t h e p r o d u c t rule, as a l r e a d y ensured. The cycle index is C , (t 1, t 2 . . . tn)/n!, since there is complete s y m m e t r y in t h e n branches.

The choices of k which r e m a i n i n v a r i a n t u n d e r a cyclic p e r m u t a t i o n of length k are exclusively of like objects, b u t b y t h e r e m a r k at t h e end of Section 2, no t w o labeled branches at the r o o t can be alike; hence Sk = r (x k, 0).

The r o o t m a y be labeled or unlabeled, hence contributes x ( l + y ) t o t h e enumeration. B y this r e m a r k a n d t h e t h e o r e m

rn (x, y) = x (1 + y) C~ (r (x, y), r (x ~, O) . . . r (x ~, O))/n!

a n d finally, b y (3) a n d (1),

r ( x , y ) = ~ . r ~ ( x , y ) = x ( l § (r(x, y ) § 2 4 7 2 4 7 2 4 7 (4) where for b r e v i t y r ( x k, O ) - - r ( x k ) . N o t i n g t h a t

r (x, 0) = r (x) = x exp (r (x)) § r (x2)/2 + ... + r (xk)/k § ...),

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which is a well-known result (P6lya [8], e q u a t i o n 1', p. 149) a n d hence a verification, e q u a t i o n (4) m a y be given t h e s y m m e t r i c a l f o r m

r (x, y) exp r (x) = (1 + y) r (x) exp r (x, y).

(4 a)

E q u a t i o n (4) constitutes the complete e n u m e r a t i o n of nonisomorphic r o o t e d trees with point labels, a n d to emphasize its i m p o r t a n c e is s u m m a r i z e d in

T H E O R E M 1. The numbers rp,n o/ rooted trees with p points, m o] which have distinct labels, are completely determined by the enumerator identity (4) with

r (x, y) = Z r~m x p yP/m!

F o r ease of evalution of these n u m b e r s it is helpful to develop some consequences of e q u a t i o n (4).

D e n o t i n g partial derivatives b y t h e usual suffix notation, e q u a t i o n (4 a) has as i m m e d i a t e consequenses:

a (x) rx (x, y) = (1 § y) ry (x, y) = r/(1 - r) (6) with r = r (x, y) a n d a (x) = r ( x ) / r ' (x) (1 - r (x)),

t h e prime denoting a derivative. Since r (x) is a p o w e r series with integral coefficients

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T H E : N U M B E R S O F L A B E L E D C O L O R E D A:ND C H R O M A T I C T R E E S 2 1 7

so is r (x)/(1 - r (x)) w h i c h is e q u a l to r (x) § r 2 (x) + .-- . I t t h e n follows b y r e c u r r e n c e f r o m its d e f i n i t i o n t h a t a (x) is also; i n d e e d b y d i r e c t c a l c u l a t i o n

a ( x ) E a n x ~ - x - x 3 - x 4 - 2 x 5 § 6 3 x 7 § s - x 9~ x 1 ~ N o t i c e t h a t t h e first half of (6) c o r r e s p o n d s to t h e r e c u r r e n c e

(1 § y) rp (y) = kr~ (y) ap+l -k (7)

k 1

w i t h t h e p r i m e d e n o t i n g a d e r i v a t i v e a n d r , (y) d e f i n e d b y t h e l a s t f o r m of (2); as- s u m i n g r p : r , (0) c o m p u t e d i n d e p e n d e n t l y ( b y e q u a t i o n (5) e.g.) t h i s seems to be t h e s i m p l e s t c o m p u t i n g f o r m u l a .

F o r c o n c r e t e n e s s i t m a y be n o t i c e d t h a t

r ( x , y ) = x ( l + y ) ~ x 2 ( l + 2 y § a ( 2 § 2 4 7 2 4 7 + x 4 (4 § 13 y + 34 y e / 2 + 64 y a / 6 § 64 y4/24) +

+ x 5 (9 + 35 y + 119 y~/2 + 326 y 3 / 6 § 625 y4/24 § 625 y5/120) § . . . . T h e s e n u m e r i c a l r e s u l t s are c o n s i s t e n t w i t h r n ~ - n " 1, which is a r e s u l t due t o Cayley. I t is i n t e r e s t i n g to see h o w (4 a) l e a d s to i t s p r o o f . M a k e t h e s u b s t i t u t i o n x = x ; x y - z in (2) a n d ( 4 a ) ; (2) m a y be r e w r i t t e n

r (x, z) = R 0 (z) + x R i (z) + ... + x p Rp (z) + ... (8) R p (z) = ~ rp+,~, m z'~/m!

m=0

a n d (4 a) b e c o m e s

r (x, z) e x p r (x) = (x ~-z) (r ( x ) / x ) exp r (x, z). (9) Since r ( x ) / x = 1 § x + . . . , i t follows f r o m (9) t h a t

r (0, z) = R 0 (z) - x e x p R 0 (z), (10)

which is P61ya's e q u a t i o n (l.c. (2.37) p. 200) f r o m w h i c h he e s t a b l i s h e s C a y l e y ' s result.

F o r c o m p l e t e n e s s i t m a y be n o t e d t h a t

x a (x) rx (x, z) = (x ~ § x z - z a (x)) r~ (x, z), ( l l ) which i m p l i e s a r e c u r r e n c e in Rp (z) a n d its d e r i v a t i v e s . A n i n s t a n c e of t h i s is

R, (z) = R0 (z),

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218 J O H N R I O R D A N

w h i c h verifies t h e i d e n t i t y r~, ~ 1 = rn~ a p p e a r i n g in t h e n u m e r i c a l r e s u l t s a n d o t h e r - wise e v i d e n t b y a s i m p l e a r g u m e n t .

T u r n i n g n o w to t h e colored ease, t h e n u m b e r of colors c m a y b e r e g a r d e d as fixed; t h e e n u m e r a t o r of r o o t e d t r e e s w i t h colored p o i n t s is t a k e n as

q (x; c) = x q l (c) + x 2 q~ (c) + . . . (12) w i t h e as p a r a m e t e r . N o t e t h a t q (x; 1 ) = r (x) in t h e n o t a t i o n a p p e a r i n g a b o v e .

Consider t h e colored r o o t e d t r e e s w i t h n b r a n c h e s a t t h e root. T h e s t o r e enu- m e r a t o r is t h a t of p l a n t e d trees, h e n c e is q(x; c), a n d Sk (x)=q(xk; c). T h e cycle i n d e x , as a b o v e , is C~ (t 1, t 2 .. . . . tn)/n! T h e r o o t m a y be colored in c ways. H e n c e b y t h e t h e o r e m a n d (1)

q (x; c) = x c e x p (q (x; c) -§ q (x2; c)/2 + ... q (x~; c)/k + . . . ) . (13) I t m a y b e h e l p f u l to s u m m a r i z e t h i s r e s u l t also in

T H E O R E M 2. The numbers qp(c) o/ rooted trees with p points, each o] which may be colored with any o/ e colors, are completely determined by the enumerator identity (13) with

q (x; c) = Z x" q~ (e).

I t m a y b e n o t e d for n u m e r i c a l e v a l u a t i o n s of t h e coefficients t h a t

xq~(x; c ) = q ( x ; c ) [ l + x q x ( x ; c)+x2q~(x2; c ) + . . . + x k q x ( x Z ; c ) + ...]. (14) T h e r e c u r r e n c e o b t a i n e d f r o m t h i s m a y be u s e d t o e v a l u a t e t h e f u n c t i o n s q~ (c) as p o l y n o m i a l s in c in t h e m a n n e r f a m i l i a r f r o m t h e e v a l u a t i o n of t h e coefficients r~

of r (x).

These p o l y n o m i a l s in t h e i r t u r n d e t e r m i n e t h e N e w t o n series

w h i c h is of i n t e r e s t b e c a u s e t h e coefficients Q~k a r e t h e n u m b e r s of r o o t e d t r e e s w i t h n p o i n t s a n d /~ specified colors. N o t e t h a t Qnl = r n 0 = rn, Qn,~ = r,,~. T h e f i r s t few results a r e

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T H E N U M B E R S O F L A B E L E D C O L O R E D A N D C H R O M A T I C T R E E S 219 F o r the c h r o m a t i c case, t a k e p (x; c) for t h e enumerator, with c again a parameter, a n d consider again t h e r o o t e d trees with n branches at t h e root. The r o o t again m a y h a v e a n y one of c colors b u t none of t h e (planted) trees at t h e root m a y h a v e t h e r o o t color on t h e p o i n t a d j a c e n t to t h e root. The n u m b e r of r o o t e d trees with a n y given r o o t color is the same as t h a t with a n y other given color.

Hence t h e store e n u m e r a t o r is ( ( c - 1 ) / c ) p (x; c) a n d it follows at once t h a t c - 1

p (x; c) = x c exp ~ - (p (x; c) + p (x2; c)/2 + ... + p (xk; c)/k + ...) (16) and, for the recurrence relations,

xpx (x; c) = p (x; c) [1 + c ~ 1 (~p~ (x; ~) + x 2 px (x2; c) + . . . x k p~ (xk; c)

+...)].

L ^c ]

The first few values of the N e w t o n series are

(;) (;)

Pl = c P8 = 4 + 9

Notice t h a t in a n o t a t i o n corresponding t o (15), P~2 = 2 r.0 (n > 1), P ~ = r~., which serve as verifications.

4. R o o t e d Trees w i t h Line Labels

The procedure of course is t h e same as above, a n d o n l y points of difference will be noticed.

F o r t h e labeled case, t a k e r* (x, y) as t h e e n u m e r a t o r b y n u m b e r of lines a n d n u m b e r of labels with r* (x, y ) = 1 § x r~ ( y ) § . . . . T h e store for the t h e o r e m n o w has e n u m e r a t o r x (1 + y ) r* (x, y) since a line at t h e r o o t m a y be labeled or not, a n d con- n e c t e d to a n y line labeled r o o t e d tree. B u t b y shifting point labels to lines f r o m t h e o u t e r points in, it is clear t h a t t h e store e n u m e r a t o r is also r (x, y). Hence t h e es.

sential relation is

x ( l + y ) r* (x, y ) = r ( x , y) (17)

which of course implies t h a t r~ (y) has a f a c t o r ( l + y ) .

F o r t h e colored case, take q* (x; c) as t h e e n u m e r a t o r with q* (x; c) = 1 + xq~ (c) + x~q * ( c ) + . . . ; then, just as above

xcq* (x; c ) = q (x; c). (18)

(10)

220 J o i ~ ttIORDAN

T h e c h r o m a t i c condition introduces essential differences. Take p* (x; c ) a s the e n u m e r a t o r for r o o t e d c h r o m a t i c trees with n lines at the r o o t with p* (x; c ) =

1 +xp'~ (c)+...,

a n d c again a p a r a m e t e r . W i t h n branches at t h e root, no two of the lines at the root of these branches, which are stems of p l a n t e d trees, m a y have the same color since t h e y have the r o o t as c o m m o n point. T a k e

g,~ (c)as

the n u m b e r of c h r o m a t i c p l a n t e d trees with c line colors, n q-1 lines, a n d a given color on the stem. T h e n the store e n u m e r a t o r in the t h e o r e m is

xg

(x; c ) - x [1 +

xg 1

( c ) + x ~g~(c) + ...]. (19) Since each p l a n t e d tree at the r o o t has a different stem color s y m m e t r y is lost;

instead the cycle index is t~ a n d since n colors for the stems m a y be chosen f r o m

Hence p * ( x ; c ) = l + ~ p* (x; c) = [l + x g (x; c)] c. (21)

r ~ : J.

On the other h a n d

g(x; c)

m a y be e n u m e r a t e d in terms of p* (x; c), since a planted tree is f o r m e d by a d d i n g a stem to a r o o t e d tree; thus exactly as a b o v e

C--n

^,

}

g (x; c) = 1 + E - - - p~ (x; c) c

= [1 §

xg

(x;

c)] c 1.

(22)

The f a c t o r

(c n ) / c = t C n l ) / t n ~ / - \ / / r ~ x

is required because n o n e o f t h e n l i n e s j o i n e d to the stem m a y have the s t e m color.

Equations (21) a n d (22) completely determine both enumerators p*(x; c ) a n d g(x; c); t h u s e.g. g(x; 1 ) = 1 a n d p * ( x , l ) = l + x , w h i l e g ( x ; 2 ) = l + x g ( x ; 2 ) = ( 1 - x ) -1 a n d p* (x; 2) - g2 (x; 2) = (1 x) -2.

To determine the polynomials p* (c) a n d their N e w t o n series defined as in (15), the following d e v e l o p m e n t is helpful. First b y differentiation of (22) it follows t h a t

gx (x; c) = I c - l + (C~22) X ~x] g2 (x; c )

(23)

a n d f r o m (21) a n d (22)

p* (x; c) = [1 + x g (x; c)] g (z: c).

(24)

These lead to t h e recurrences

(11)

THE N U M B E R S OF L A B E L E D COLORED AND CHROMATIC T R E E S 221 2 n gn (c) = [(n + 1) c - 2 n] g~_ 1, 2 (c) t (25) 2 n p * ( c ) = ( n § l ) c g ~ _ l . 2 ( c ) )

with gn2 (c) defined b y g2 (x; c) = 1 § ~ x ~ g~2 (c) n = l

or b y its consequence

gn2 (C) - - g n - 1 (C) g l (C) § gn 2 (C) ~2 (C) -] .. . . § g l (C) gn 1 (C),

These lead to a serial c o m p u t a t i o n , the first few results of which in N e w t o n series f o r m are

p ~ ' = 3 2 p ~ = 6 2 \ 3 ] § 1296

5. T r e e s

F o r unlabeled trees, e n u m e r a t e d b y n u m b e r of points b y t (x) = t 1 x ~ t 2 x 2 ~i ... , it is k n o w n (Otter [7]) t h a t

1 2 1

t (x) = r (x) - ~ r (x) § ~ r (x 2) (26)

where r (x) is the corresponding e n u m e r a t o r for r o o t e d trees. PSlya [8] has p r o v e d an equivalent, t h o u g h less compact, result b y a procedure which is easily a d a p t e d to t h e m a r k e d cases.

Briefly this consists of dividing t h e e n u m e r a t i o n b y considering separately t h e centroidal a n d bicentroidal trees defined above, a n d relating these e n u m e r a t i o n s t o those for r o o t e d trees, following the obvious suggestion of the pictures of these trees.

As a r e m a i n d e r for the reader, note again t h a t bicentroidal trees with an o d d n u m b e r of points are impossible, while for an even n u m b e r ( p = 2 q ) t h e branches at each centroid, excluding the one t h r o u g h t h e other, h a v e in total q points; the picture is of two r o o t e d trees joined b y a line. The centroidal trees with p points h a v e branches at t h e centroid h a v i n g at m o s t [ ( p - 1 ) / 2 ] points.

F o r labeled points t a k e t (x, y) as the e n u m e r a t o r ; as before P

t (x, y) = x t 1 (y) § x 2 t~ (y) § . . . . ~ x ~ ~ tpm y m / m ! (27) m~0

(12)

222 Also,

J O H N R I O R D A N

for t h e division into centroidal a n d bicentroidal trees, write

t (x, y) = t' (x, y) + t 1' (x, y).

(2s)

B y the r e m a r k s above a n d b y t h e t h e o r e m t h e bicentroidal trees are completely e n u m e r a t e d b y

t2q (y) = ~ [r2q (y) § rq (0]

t~'q+ 1 (y) = 0.

F o r t h e centroidal trees, those r o o t e d trees h a v i n g m o r e t h a n [ ( p - 1 ) / 2 ] points in a n y b r a n c h at t h e r o o t m u s t be s u b t r a c t e d f r o m t h e t o t a l a n d after some simpli- fication it t u r n s o u t t h a t

t~q (y) = r2q (y) - r2, 1 (Y) rl (Y) - r2,-2 (y) r2 (y) . . . rq (y) 2

t' ~q+l (Y) = r~q+l (y) - ruq (y) r 1 (y) - r 2 q _ i (y) r~ (y) . . . r q + l (y) rq (y).

S u m m i n g these on p results in

I S 1

t (x, y) = r (x, y) - ~ r (x, y) § ~ r (x2). (29)

F o r y = 0, this is Otter's f o r m u l a (equation (26) above).

W i t h a (x) as in (6) a n d a prime d e n o t i n g a derivative t h e results corresponding to (6) are

a (x) t~ (x, y) = r (x, y) § x a (x) r' (x ~) (30)

(1 § y) ty (x, y) = r (x, y ) = a (x) tx (x, y) - x a (x) r" (x2).

B y t h e last of these t~ (x, 0 ) = r (x),

which is t h e same as tpl = rr, i.e. t h e n u m b e r of trees with just one label equals t h e n u m b e r of r o o t e d trees, a n o t h e r verification.

Finally with z = x y as in (8) a n d

t (x, z) = T O (z) + x T 1 (z) + . . . (31)

w i t h T~ (z) = Z t~+m. m zm / m !

it follows f r o m ( x + z) tz (x, z) = r (x, z), (32)

t h a t T'n 1 (Z) § z T n ( z ) = R~ (z), (33)

(13)

T H E N U M B E R S OF L A B E L E D C O L O R E D A N D C H R O M A T I C T R E E S 223 which entails in particular z To (z)=R 0 (z) or passing to coefficients, n tnn = rnn = n ~-1, so tnn = n n-u, a n o t h e r of Cayley's results, and hence a verification.

F o r trees with colored points there is no essential change from the unlabeled case and if u (x; c) is the e n u m e r a t o r

1 q 2 ( x ; c ) + ~ (x~;c) (34)

u (x; c) = q (x; c) - ~ q

F o r point chromatic trees take v (x; c) as t h e enumerator. T h e bicentroidal trees are required to have different colors a t t h e two centroids, hence

(c) = (c).

A similar a d j u s t m e n t is required for centroidal trees with the final result t h a t l c - 1

- - - - p ~ (x; c) ( 3 5 )

v (x; c) = p (x; c) 2 c

F o r line marks, the same notational procedure as for rooted trees is followed:

a superscript star denotes t h e line m a r k e d case.

F o r line labeled trees t* (x, y) is the e n u m e r a t o r , b u t for convenience t* (x, y) and t~ (x, y) are the e n u m e r a t o r s of centroidal and bicentroidal trees, respectively. R e m e m - bering t h a t x is the variable for n u m b e r of lines, the results for bicentroidal trees are

t* 8. ~q (Y) = 0

t~. 2q+1 (y) = (1 + y) [(r* (y))~ + r* (0)]/2.

Adjusting similarly for centroidal trees, it is found t h a t

t* (x, y) = r* (x, y) - ~ x (1 + y) (r* (x, y)) § ~ x (1 § y) r* (x~). 1 (36)

I t is w o r t h noting t h a t

1 ~ 1

x ( l § (x, y ) = r ( x , y ) - ~ r (x, y ) § § 2)

= t (x, y) § (y § y V 2 ) r (x~). (37) F o r line colored trees, u* (x, c) is the e n u m e r a t o r and b y an a r g u m e n t like t h a t above

1 1

u* (x; c ) = q * (x; c ) - ~ x c (q* (x; c))~ + ~ xcq* (x~; c) (38)

(14)

2 2 4 J O H N RIORDAN

] q2 (X; C) C x c u* (x; c) = q (x; c) - ~ + ~ q (x~; c)

C - - 1

= u (x; c) + ~ - q (x~; c).

F o r line c h r o m a t i c trees, v* (x; c) is the e n u m e r a t o r a n d

1 ~ 1

v*(x; c ) = p * ( x ;

c)-2xcg

^(x;

c)+2xcg(x~;

^c)

with g (x; c) the e n u m e r a t o r defined b y (19).

(39)

(40)

§ v (x2; c)/2 + . . . ) , (45)

x (1 + y) q* (x, y) = ~ (x, y), x ( l + y ) ~ * ( x , y ) = ~ ( x , y ) .

(43) (44)

(46)

(47)

(48)

v (x; c) = z (x; c) - u s (x; c).

F o r the p o i n t c h r o m a t i c case

v(x; c ) = x c exp 2 C c 1~ (v (x; c)

(1) (x; c) = v (x; c) - c - 1 v2 (x; c).

v F o r line labels

6. Oriented R o o t e d a n d Free Trees

These differ f r o m the trees a b o v e only b y h a v i n g each line o r i e n t e d in one of t h e t w o possible directions. As is to be expected the e n u m e r a t i o n s differ only in m i n o r details, so this section is m a i n l y a c o m p e n d i u m . Greek letters are used for t h e enumerators, ~, ~r, a n d v for r o o t e d trees a n d 3, v, a n d (I) for trees, with a star as before distinguishing t h e line labeled cases.

F o r the point labeled case the results are

e (x, y ) = x (1 ~ y ) e x p 2 (e (x, y ) + ~ (x~)/2 + . - . + e ( x k ) / k + ' ' ' ) , (41)

T (x, y) = e (x, y) - ~2 (x, y). (42)

F o r the p o i n t colored case

(x; c) - x c exp 2 (n (x; c) ~ ~r (x2; c)/2 + . . . + :~ (xk; c ) / k ~ .... ),

(15)

THE NUMBERS OF LABELED COLORED AND CHROMATIC TREES F o r l i n e colors

xc~*

(x;

c)=~

(x; c), x c v* (x; c) = v (x; c).

:For t h e l i n e c h r o m a t i c case

v* (x; c) = p* (2 x; c), (I)* (x; c) = v* (2 x; c )

225

(49)

(50)

(51) (52) T h e p o i n t l a b e l e d case h a s a d e v e l o p m e n t like its u n o r i e n t e d c o r r e s p o n d e n t w h i c h is o m i t t e d t o s a v e space; i t h a s b e e n u s e d t o o b t a i n t h e n u m b e r s a p p e a r i n g i n

e (x, y) = x (1 + y) + x 2 (2 + 4 y + 4 y~/2) + x a (7 + 19 y + 36 y 2 / 2 + 36 y 3 / 6 ) + + x 4 (26 + 94 y + 264 y 2 / 2 + 512 y a / 6 + 512 y4/24) § ...

T (x, y) = x (1 + y) + x 2 (1 + 2 y + 2 y e / 2 ) + X a (3 + 7 y + 12 y 2 / 2 + 12 yS/6) + + x 4 (8 + 26 y + 68 y 2 / 2 + ] 2 8

ya/6 ~-

128 y4/24) + . - - .

References

[1]. E . T. BELL, E u l e r A l g e b r a . Trans. A m . Math. Soc., 25 (1923), 135-154.

[2]. L. CARLITZ (~5 J . RIORDAN, T h e n m n b e r of labeled t w o - t e r m i n a l series-parallel n e t w o r k s . Duke Math. Journal, 23 (1956), 435-446.

[3]. A. CAYLEY, Collected Mathematical Papers. C a m b r i d g e , 1889-1897; Vol. 3, 242 246; Vol. 9, 202-204, 427-460; Vol. 11, 365-367; Vol. 13, 26-28.

[4]. G. W. FORD & G. E . UHLENBECK, C o m b i n a t o r i a l p r o b l e m s in t h e t b e o r y of graphs. I.

Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 122-128.

[5]. R . M. FOSTER, T h e n u m b e r of series-parallel n e t w o r k s . Proceedings o/ the Ir~ternational Congress o/Mathematicians (1950), Vol. I, 646.

[6]. F. HARA~Y, T h e n u m b e r of linear, directed, rooted, a n d c o n n e c t e d graphs. Trans. A m . Math. Soc., 78 (1955), 445-463.

[7]. R . OTTER, T h e n u m b e r of trees. A n n . o] Math., 49 (1948), 583-599.

[8]. G. P61ya, K o m b i n a t o r i s e h e A n z a h l b e s t i m m u n g e n ftir G r u p p e n , G r a p h e n u n d c h e m i s c h e V e r b i n d u n g e n . Acta. Math., 68 (1937), 145-253.

15-563804. Acta ma~hematica. 97. Imprim5 le 7 aofit 1957.