Discontinuities in the asymptotics of plane trees

Discontinuities in the asymptotics of plane trees

Citation for published version (APA):

Bruijn, de, N. G. (1977). Discontinuities in the asymptotics of plane trees. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7707). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1977

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_{EINDHOVEN UNIVERSITY OF TECHNOLOGY}

Department of Mathematics

Memorandum 1977-07 Issued July 1977.

Discontinuities in the asymptotics of plane trees.

by N.G. de Bruijn. University of Technology Department of Mathematics P.O.Box 513, Eindhoven. The Netherlands.

Discontinuities in the aSymptotics of plane trees.

by

N.G. de Bruijn.

I.Introduction. Recently D.A.Klarner (Binghamton, N.Y.) proposed a question on the radius of convergence of the generating series for certain classes of trees. If T is a set of plane trees he considers the set SeT) of all those plane trees of which no subtree belongs to T. Let peT) be the radius of convergence of the generating

. 00

function of SeT). His question was: if T

t C T2, ••• , Un=l Tn

=

T, does it follow that peT ) + peT)?

n

There are various possibilities for the definition of "subtree". In this note we take a simple definition (different form Klarner's), we study the p's and show

that not necessariiy p (T) + peT) (section vi (ii». n

2. Definitions. Here we explain the various notions used in section 1. Rather than producing a formal definition of the kind of trees we consider, we give a list of the first few; under each tree we write the number of vertices:

v

v

Y

2 3 3 4 4 4 4 4 5

(Some people would call these "planted plane trees with the roots cut off", or just "planted plane trees", others like to give a recursive definition and say: a tree

4It

is a root plus a (possibly empty) sequence of edges leaving from it, and on each

edge we have grown a tree).

A subtree of a tree is obtained by taking a vertex of that tree plus everything above it. So the different subtrees of

are

V,

If s is a subtree of t, and s ~ t then s is called a proper subtree of t. If t is a tree, then net) denotes the number of vertices.

,

n(t).

x

The radius of convergence is denoted by p(T) (possibly p(T)=oo).

2.

If T is a set of trees, then S(T) is the set of all trees s which do not have any t € T as a subtree.

3.The partial order. We denote the set of all trees by W. tn W we take the partial order relation s: we write s ~ tiff s is a subtree of t. If T c W,

and t E T, then t is called a minimum of T i f s € T and s ~ t imply s := t. I f

min(T) is the set of minima of T, then min(T) c T, and to every t E T there

is at least one s € min(T) with s ~ t. This easily follows from the fact that

in every descending sequence of trees t} ~ t2 ~ t3 ~ .•• the tn is constant

from a certain index onward.

It is clear that S(T) = S(min(T».

If A c W, and if A has the property that for all sEA, t € W with t ~ s

we have tEA then Ais called conservative. It is easy to see that A is con-servative if and only if there is a T (T c W) with A = S(T). Moreover, if A is conservative we have iA = S(W _\ A).

A subset B of W can be written in the form B = min(T) with some T, T c

w,

i f and only i f B is an independence set, Le. i f never s ~ t with SE B, t € B,

s :/: t.

4. The g,enerating functions.

Theorem. If T is an independence set, we have (coefficientwise)

(4.1)

Proof. As S(T) and T are disjoint, we have, on the left, the generating function of S(T) u T. This can be described as the set of all trees of which no proper subtree lies in T. We can partition S(T) u T in a second way, where one part consits of the one-vertex tree only, and, for n

=

1,2, ••. , the n-th part consists of all trees where n edges leave the root and where on the end-point of each edge there grows one of the trees of S(T). This partition corres-ponds to the right-hand side.

S.Convergence and analyticitI' Formula (4.1) was intended as a formal relation between power series with positive coefficients. But as the series on the left is dominated by fW(x) , and since it is not hard to show that the number of trees with n vertices is s 4n, the series in (4.1) converge at least for Ixl <

!.

I

3.

A theorem of Pringsheim says that i f p is the radius of convergence of

a power series with non-negative coefficients, and if 0 < p < 00, then the sum

of the series has a singularity at x=p. Now let PI and P2 be the radii of

fS(T) and f

T, respectively; so 0 < PI ~ ~, 0 < P

z

~ 00. For a small positive

value of x we can solve fS(T) from (4.1):

(5.1)

Observing fT(x) on the interval 0 ~ x ~ P

Z' we see tbree cases:

(i) (I+fT(X»Z > 4x (0

~

x < P

Z)' Now fS(T) is analytic on that

segment, and we infer that PI ~ P

2, (ii) (1+f

T(x»2 > 4x (0

~

x < c) and (J+fT(C»Z

=

4c,

2fT'(c)(l+fT(c»

=

4 for some c with 0 < c < P

2, We can now argue that fT(x)

is still analytic at c. Noting that fT'(x) has non-negative coefficients we

2

derive that (I + fT(x» > 4x (c < x < P2)' Our conclusion is again that

PI ~ PZ'

(iii) (l+f_T(x»2 > 4x for some interval 0

~

x < c with 0 < c < P

Z' and

2

(I + fT(x» < 4x for all x in some interval c < x < ct. Now fS(T) has its

first singularity at c, whence p

1=c.

6.Applications. (i) If T is taken to be empty then SeT)

=

W, We are in case

(iii) of section 5, with c

= 1,

and by (5.1), we get the well-known formula

f (x)

=

i -

~

(1-4x)

~

=

W

(ii) We define the trees

a

=

I

t ' a 2

=

v,

n x • a ==

'V

3

,

.".

,

and the sets Tn == {at' ••• ,an}' T = {at ,a

Z" " }. For every n, the set Tn

is an independence set. We have

2 3 Z - 1

fT(x)

=

x + x + •••

=

x (I-x) ,

and the funny identity

So P

2=1 and PI ~ 1 (section 5, case (ii». Actually we get PI

=

00, since

2

fS(T)(x)

=

x by (5.1). The number c where (l+f

r

,

solution of x2-3x+l=O, ratio number ~(-1+15».

i.e. ~(3-15)=·381966 (it is the square of the golden

4.

If we replace x2 + x3 + ••• by its truncation x2 + x 3 + ••• + x , n it is easy to see that we get case (iii) of section 5, with a value of c that tends to ~(3-15) as n tends to infinity. So with the notation of section 1 we have

p (T ) -+

H3-1S),

n peT)

=

00 •

It is not hard to see what the elements of SeT ) look like. Apart from

n .

the one-vertex tree they might be called "brushy trees" : we get them from an arbitrary tree if we grow on each end-point a new tree, taken from the collection a_n+_l,a

n+2,... • The set SeT), however, consists of the one-vertex tree only.

(iii) The following example just serves as a further illustration to the contents of sections 3 and 4. We start from a conservative set A, viz. A

=

=

{b

1,b2, ••• }, where

b =

2 b 3

=

By section 3 we have A = SeW \ A)

=

S(min(W \ A». I f T is defined as min(W \ A) t

we have by (4.1) ( -X- x -2 fT x)

=

t-x - x(I - I-x) 3 x = ""(-:-I--x"""<')-:(-:-1-~2:-x"""")

What is T1 I f t € T then t is minimal in W \ A, i.e. t itself is not in A but

every proper subtree of t is in A. So t looks like this

with at least 2 edges leaving the root. The number of such trees with n+l points equals the number of solutions of ul+ ••. +~=n in positive integers

n-l