Table Of ContentLIMIT LAWS FOR EMBEDDED TREES.
APPLICATIONS TO THE INTEGRATED SUPERBROWNIAN
EXCURSION
MIREILLEBOUSQUET-MÉLOU
5
0
0 Abstra
0t. Westudythreefamiliesoflabelledplanetree0s,.1Inal−l1thesetrees,therootis
2 labelled ,andthelabelsoftwoadja
entnodesdi(cid:27)erby or .
n j ∈ONnepart ofthepaper isdevoted toenumerative results. For ea
h family,and for all
, we obtain
losed form expressions for the following three generating fun
tions:
a j
thegenerating fun
tionoftrees havingnolabellargerthan ;the(bivariate) generating
J j
fun
tion of trees,
ounted by the number of edges and the number of nodes labelled ;
8 and (cid:28)nally the (bivariate) generating fun
tion oftrees,
ounted by the number of edges
j
1 and the number ofnodes labelledat least . Strangely enough, allthese series turn out
tobealgebrai
, butwehaveno
ombinatorialintuitionforthisalgebrai
ity.
] Theother part ofthepaper isdevotedtoderivinglimitlawsfromtheseenumerative
O n
results. In ea
h of our familiesof trees, we endow the trees of size with the uniform
Mn
C distribution,andXstnu(djy)thefollowingrandomvariablesj: ,tXhne+l(ajr)gestlabelo
urringin
. a(randomj )tree; ,thenumberofnodeslabelled ;and ,thenumberofnodes
h labelled ormore. Weobtainlimitlawsfors
aledversions oftheserandomvariables.
at Finally,wetranslatetheabovelimitresultsintostatementsdealingwiththeintegrated
m superBrownian ex
ursion (ISE). In parti
ular, we des
ribe the law of the supremum of
itssupport (thusre
overing someearlierresultsobtained byDelmas),andthelawofits
[ distribution fun
tion at a given point. We also
onje
ture the law of its density (at a
givenpoint).
1
v
6
6
2
1 1. Introdu
tion
0
5 We study in this paper three families of labelled plane trees. In all these trees, the root
0 0,1 1
0 is labelled , and the labels of two adja
ent nodes di(cid:27)er by or − .
/
h More pre
isely, the (cid:28)rst family we
onsider is th1e set of plane trees, and the in
rements
t of the labels along edges are
onstrained to be ± . In the
losely related se
ond family,
a 0, 1
m these in
rements
an be ± . The third family is a bit di(cid:27)erent. It is simply the set of
(in
omplete) binary trees, in whi
h the nodes are labelled in a deterministi
way: the label
:
v of a node is the di(cid:27)eren
e between the number of right steps and the number of left steps
i o
urringinthepaththatyieldsfromtheroottothenodeunder
onsideration. SeeFigure1
X
for an illustration. We
all this labelling the natural labelling of the binary tree. Note that
r
a the label of ea
h node is simply its abs
issa, if we draw the tree in the plane in su
h a way
the right(resp. left) sonofanode liesone unit to the right(resp. left) of its father. Forthis
reason, we will sometimes
all these labelled binary trees naturally embedded binary trees.
Moregenerally,forany planelabelledtree,wemay
onsiderthatthelabel ofea
hnodetells
Z
where to embed it in ; hen
e the title of the paper.
n
Inea
hofthesethreefamilies,weendowthesetoftreeshavingagivensize(say, edges)
with the uniform distribution. We address (via generating fun
tions) the following three
questions:
Date:January 17,2005.
MBMwaspartiallysupportedbytheEuropeanCommission'sIHRPProgramme,grantHPRN-CT-2001-
00272,(cid:16)Algebrai
Combinatori
sinEurope(cid:17).
1
2 MIREILLEBOUSQUET-MÉLOU
1 1
2 0 −2 1 0 −1 0 2
1 1 −1 1 1 0 −1
1
0
0 0
1
Figure 1. A labelled plane tree with in
rements ± . (cid:22) A labelled tree
0, 1
with in
rements ± . (cid:22) A naturally embedded binary tree.
(1) What is the maximal label that o
urs in the tree? This label is in fa
t a random
M M /n1/4
n n
variable . Weprovethat
onvergesindistributiontoarandomvariable
N
having a density. We give this density expli
itly. We also
ompute the moments
N M /n1/4 N
n
of and prove the
onvergen
e of the moments of to those of .
j X (j)
n
(2) How many nodes of the tree have label ? Let denote the
orresponding
j n
randomvariable. If is(cid:28)xed,and goestoin(cid:28)nity,thentheanswertothisquestion
j j Z X (j)/n3/4
n
isindependent of . Weprovethatforany ∈ , thevariable
onverges
cT 1/2 c
−
in distribution to , where is a
onstant depending on whi
h family of trees
T 2/3
we
onsider, and follows a unilateral stable law of parameter .
n1/4
Given that the maximal label grows like , we get a better insight on the
n λn1/4
label distribution by asking how many nodes in a tree of size have label ⌊ ⌋.
λ R X ( λn1/4 )/n3/4
n
We prove that, for any ∈ , the random variable ⌊ ⌋
onverges
Y(λ)
in distribution to a limit variable . This variable admits a Lapla
e transform,
whi
h we give expli
itly. The
onvergen
e of the Lapla
e transform, and of the
moments, hold as well. We say we have obtained a lo
al limit law for embedded
trees, be
ause we look at one value of the labels only.
X+(j)
(3) Finally, we also obtain a global limit law by studying the variable n that gives
j X+(0)/n
thenumberofnodeshavinglabel atleast. Remarkably,weprovethat n ,the
(normalized)numberofnodeshavinganon-negativelabel,
onvergestotheuniform
[0,1] λ R X+(λn1/4)/n
distributionon . Moregenerally,for ∈ ,thevariable n
onverges
Y+(λ)
indistributiontoavariable . ThisvariableadmitsaLapla
etransform,whi
h
wegiveexpli
itly. On
eagain, the
onvergen
eofthe Lapla
etransform, and ofthe
moments, hold as well.
N Y(λ) Y+(λ)
The laws of , and naturallydepend on whi
h family of treeswe
onsider,but
only by a simple normalization fa
tor.
1.1. Embedded trees and the integrated superBrownian ex
ursion
Why should one study su
h labelled trees?
The (cid:28)rst two
lasses of trees we
onsider have a
lose
onne
tion with
ertain families of
n
planar maps [6, 8, 11℄. In parti
ular, the diameter of a random quadrangulation having
fa
es is distributed like the largest label in non-negative random trees of our se
ond family.
n1/4 (M m )n 1/4
n n −
Moreover, on
e s
aled by , this diameter has the same limit law as − ,
M m
n n
where (resp. ) is the largest (resp. smallest) label o
urring in a random tree of our
se
ond family [8℄.
The third
lass we study is the good old family of binary trees, and this may su(cid:30)
e to
motivate its study! More seriously, the three questions addressed above have, for binary
M
n
trees, a natural geometri
formulation. The random variable (the maximum label) tells
usaboutthe(cid:16)truewidth(cid:17) ofabinarytree(asopposedtothemaximalnumberofnodeslying
√n X (j)
n
at the samelevel, whi
h isknownto growlike ). More generally, the variables tell
LIMIT LAWS FOR EMBEDDED TREES 3
2 2 4 2 1 1
4 2
3 3
2 4
1 2
0 1
−2 −1 0 1 2 3
[1,2,4,3,2]
Figure2. An(in
omplete)binarytreehavinghorizontalpro(cid:28)le
[2,2,4;2,1,1]
and verti
al pro(cid:28)le .
us about the verti
al pro(cid:28)le of the tree (as opposed to the horizontal pro(cid:28)le whi
h des
ribes
the repartition of nodes by level [13℄). See Figure 2.
We may also invoke an a posteriori justi(cid:28)
ation to the study of these trees: the form of
the generating fun
tions we obtain is remarkable,whatever family of trees we
onsider, and
suggests that there must be some beautiful hidden
ombinatori
s in these problems, whi
h
should be explored further.
However, the main motivation for this work is the
onne
tion between embedded trees
and theintegrated superBrownian ex
ursion (ISE). Chooseoneof thethreefamiliesoftrees,
R
and
onsider the following random probability distribution on :
1
µ = X (j)δ ,
n n+1 n cjn−1/4 (1)
j Z
X∈
X (j) j δ x
n x
where isthe(random)numberofnodeslabelled , denotestheDira
measureat ,
c √2 √3 1
and the
onstant equals forthe (cid:28)rst family, forthese
ondoneand forthefamily
µ
n
of binary trees. Then is known to
onverge weakly to a limiting random probability
µ
n
distribution
alled the ISE [1, 23, 22, 20℄. See Figure 3 for simulations of .
Our limit results provide some information about the law of the ISE. For instan
e, we
cM n 1/4 µ
n − n
provethat , the largest point having a positive weight under ,
onvergesin law
200 200 200 200
150 150 150 150
100 100 100 100
50 50 50 50
–15 –10 –5 0 5 10 15 20 –15 –10 –5 0 5 10 15 20 –15 –10 –5 0 5 10 15 20 –15 –10 –5 0 5 10 15 20
200 200 200 200
150 150 150 150
100 100 100 100
50 50 50 50
–15 –10 –5 0 5 10 15 20 –15 –10 –5 0 5 10 15 20 –15 –10 –5 0 5 10 15 20 –15 –10 –5 0 5 10 15 20
X (j) j n=1000
n
Figure 3. The plot of vs. for random binary trees with nodes.
4 MIREILLEBOUSQUET-MÉLOU
N
to ise, the supremum of the support of the ISE. We denote this by
cMnn−1/4 d N .
→ ise
M n 1/4
n −
The results we obtain for the limit law of thus translate into expressions of the
moments, distribution fun
tion and density of the supremum of the ISE. Note that the
moments were already obtained by Delmas [12℄. Our se
ond limit result deals with the
X ( λn1/4 )
n
random variables ⌊ ⌋ . Observe that
1
µ (cλ cn 1/4,cλ]= X ( λn1/4 ).
n − n
− n+1 ⌊ ⌋ (2)
Y(λ)
This leads us to
onje
ture that the random variable involved in our lo
al limit law
satis(cid:28)es d
Y(λ)=cf (cλ)
ise (3)
f
where ise is the (random) density of the ISE. Similarly,
1
µ [cλ,+ )= X+( λn1/4 ),
n ∞ n+1 n ⌈ ⌉
Y+(λ)
and we prove that the random variable involved in our global limit law satis(cid:28)es
Y+(λ)=d g (cλ)
ise
g
where iseYis(tλh)e(ranYd+om(λ))taildistributionfun
tionoftheISE.Theresultsweobtainfabo(λu)t
thelawsof and thustranslateintoformulasfortheLapla
etransformsof ise
g (λ) f (λ)
and ise (the formufla for ise being
onje
tural). Y(λ/c)/c
Our
onje
ture on ise is naturally supported by the fa
t that the law of is
independent of the tree family we start from. This is one of the reasons why we
onsider as
manyasthreefamiliesoftrees. Theotherreasonsinvolvethe
onne
tionswithplanarmaps,
the remarkable form of the generating fun
tions we obtain, and our unshakeable interest in
binary trees. The details of the
al
ulations are only given for the (cid:28)rst of the three families
(Se
tions 2 to 5), while the results are merely stated for the other two families (Se
tion 6).
Let us (cid:28)nally mention that the moments of the
enter of mass of the ISE have re
ently
been determined by two di(cid:27)erent approa
hes[7, 19℄. In ourdis
rete setting, this boils down
to studying the
onvergen
e of the variable
1
jX (j).
n5/4 n
j Z
X∈
1.2. Overview of the paper
The starting point of our approa
h is a series of exa
t enumerative results dealing with
1
our (cid:28)rst
lass of trees: plane trees in whi
h the labels of adja
ent nodes di(cid:27)er by ± . These
results are gathered in the next se
tion. We obtain for instan
e an expli
it expression for
the bivariate generating fun
tion of labelled trees,
ounted by the number of edges and the
j j
number of nodes labelled (for (cid:28)xed). This se
tion in
ludes, and owes a lot to, some
resultsre
entlyobtained by Bouttier, Di Fran
es
oand Guitter [5,6℄on theenumerationof
j
trees having no label greater than . This part of our work raises a number of
hallenging
ombinatorialquestions(cid:22) why aretheseexpressionssosimple? (cid:22) whi
harenotaddressed
in this paper.
M X ( λn1/4 ) X+(λn1/4)
The limit behaviours of the random variables n, n ⌊ ⌋ and n are re-
spe
tively established in the next three se
tions (Se
tions 3 to 5). The main te
hnique that
we use is the (cid:16)analysis of singularities(cid:17) of Flajolet and Odlyzko [17℄. It permits to extra
t
the asymptoti
behaviour of the
oe(cid:30)
ients of a generating fun
tion. This te
hnique has
already proved useful in numerous o
asions, in parti
ular for proving limit theorems that
aresimilarin(cid:29)avourtotheonesobtainedinthispaper: thesetheoremsdealwiththeheight
of simply generated trees and their pro(cid:28)le, whi
h are known to be related to the height of
LIMIT LAWS FOR EMBEDDED TREES 5
the Brownian ex
ursion and its lo
al time [16, 13℄. This te
hnique is
arefully exempli(cid:28)ed
in Se
tion 3 (whi
h is devoted to the maximal label) before the more di(cid:30)
ult questions of
the lo
al and global limit laws are atta
ked (Se
tions 4 and 5).
Finally, two other families of trees are brie(cid:29)y studied in Se
tion 6: trees with in
rements
0, 1
± and naturally embedded binary trees. The emphasis is put on their enumerative
properties, whi
h turn out to be as remarkable and surprising as those of our (cid:28)rst family of
trees. The limit laws we obtain are (up to a s
alar) the same as for the (cid:28)rst family.
Let us
on
lude with some notation and a few de(cid:28)nitions on formal power series and
K K[t] t
generating fun
tions. Let be a (cid:28)eld. We denote by the ring of polynomials in with
K K(t) t K
oe(cid:30)
ients in , and by the (cid:28)eld of rational fun
tions in with
oe(cid:30)
ients in . We
K[[t]] t K A(t) K[[t]]
denote by the ring of formal power series in with
oe(cid:30)
ients in . If ∈
n N [tn]A(t) tn A(t) A(t)
and ∈ , the notation stands for the
oe(cid:30)
ient of in . The series
K(t)
is said to be algebrai
over if it satis(cid:28)es a non-trivial polynomial equation of the form
P(t,A(t)) = 0 P K
, where is a bivariate polynomial with
oe(cid:30)
ients in . In this
ase, the
A(t) P
degree of is the smallest possible degree of (in its se
ond variable).
Let A be a set of dis
rete obje
ts, equipped with a size that takes nonnegative integer
n N n
values. Assume that for all ∈ , the number of obje
ts of A of size is (cid:28)nite, and denote
a
n
this number by . The generating fun
tion of the obje
ts of A,
ounted by their size, is the
formal power series
A(t)= a tn.
n
n 0
X≥
The above notions generalize in a straightforward way to multivariate power series. Su
h
series arise naturally when enumerating obje
ts a
ording to several parameters.
2. Enumerative results
We
onsider in this se
tion (and in the three following ones) our (cid:28)rst family of labelled
0 1
plane trees: the root is labelled , and the labels of two adja
ent nodes di(cid:27)er by ± .
2.1. Trees with small labels
The (cid:28)rst enumerative problem we address has already been studied by Bouttier, Di
j N
Fran
es
o and Guitter [5, 6℄. It deals with the largest label o
urring in a tree. For ∈ ,
T T (t)
j j
let ≡ be the generating fun
tion of labelled trees in whi
h all labels are less than
j t T T(t)
or equal to . The indeterminate keeps tra
k of the number of edges. Let ≡ be the
T T
j
generating fun
tion of all labelled trees. Clearly,
onverges to (in the spa
e of formal
t j
power seriesin ) as goesto in(cid:28)nity. It is very easyto des
ribe an in(cid:28)nite set of equations
T
j
that
ompletely de(cid:28)nes the
olle
tion of series .
T
Lemma 1. The series satis(cid:28)es
T =1+2tT2.
(4)
j 0
More generally, for ≥ ,
T =1+t(T +T )T
j j 1 j+1 j
−
T =0 j <0
j
while for .
Proof. The two ingredients of the proof will be useful for the other enumerative problems
k j k T
j
weaddressbelow. Firstly,repla
ingea
hlabel by − showsthat isalsothegenerating
j
fun
tionoftreesrootedat andhavingonlynon-negativelabels(wesaythatatreeisrooted
j j
at if its root has label ). Se
ondly,
onsidersu
h a tree and assumeit is not redu
ed to a
single node. The root has a leftmost
hild, whi
h is the root of a labelled subtree, rooted at
j 1
± and havingonly non-negativelabels. Deleting this subtree leavesasmallertree rooted
j
at , having only non-negativelabels (see Figure 4). The result follows.
6 MIREILLEBOUSQUET-MÉLOU
T
T j±1 T
j j
= +
j 1
j
±
j j
Figure 4. The de
omposition of plane labelled trees.
T
The above lemma shows that the series ,
ounting labelled trees by edges, is algebrai
,
and the short proof we have given provides a simple
ombinatorial explanation for this
T
j
property. What is far less
lear (cid:22) but nevertheless true (cid:22) is that ea
h of the series is
algebrai
too, as stated in the proposition below, whi
h we borrow from [5, 6℄. These series
T T(t)
will be expressed in terms of the series ≡ and of the unique formal power series
Z Z(t) 0
≡ , with
onstant term , satisfying
(1+Z)4
Z =t .
1+Z2 (5)
T Z
Observe that and are related by:
(1+Z)2
T = .
1+Z2 (6)
T T (t)
j j
Proposition2 (Trees with smalllabels [5,6℄). Let ≡ be the generatingfun
tion
j T 2
j
of trees having no label greater than . Then is algebrai
of degree (at most) . In
parti
ular,
T =1 11t t2+4t(3+2t)T 16t2T 2.
0 0 0
− − −
j 1
Moreover, for all ≥− ,
(1 Zj+1)(1 Zj+5)
T =T − − ,
j (1 Zj+2)(1 Zj+4) (7)
− −
Z Z(t)
where ≡ is given by (5).
T
j
Proof. Itisveryeasyto
he
k,using(5(cid:21)6),thattheabovevaluesof satisfythere
urren
e
T = 0
1
relation of Lemma 1 and the initial
ondition − . How to dis
over su
h a formula
T
j
is another story, whi
h is told in [5℄. The remarkable produ
t form of still awaits a
ombinatorial explanation.
T T Z j = 0
0
The equation satis(cid:28)ed by is obtained by eliminating and from the
ase
j T
j
of (7). Then an indu
tion of , based on Lemma 1, implies that ea
h is quadrati
(at
Q(t)
most) over .
Remarks
T Q(t) Z
1. The produ
t form (7),
ombined with the fa
ts that is quadrati
over and is
Q(T) T Q(t) 4
j
quadrati
over ,showsthat belongstoanextensionof ofdegree . Thisistrue,
T Q(t)
j
but not optimal, sin
e is a
tually quadrati
over . Hen
e this produ
t form does not
T
j
give the best possible information on the degree of .
T
0
2. The trees
ounted by (equivalently, the trees having only non-negative labels) are
knowntobeinbije
tionwith
ertainplanarmaps
alledEuleriantriangulations[6℄. Through
this bije
tion, the number of edges of the tree is sent to the number of bla
k fa
es of the
triangulation. These triangulations are nothing but the dual maps of the bi
ubi
(that is,
LIMIT LAWS FOR EMBEDDED TREES 7
bipartite and trivalent) maps, whi
h were (cid:28)rst enumerated by Tutte [26℄. In parti
ular, the
T (t)
0
oe(cid:30)
ients of are remarkably simple:
(1 8t)3/2 1+12t+8t2 3.2n 1 2n
T (t)= − − =1+ − tn.
0 32t2 (n+1)(n+2) n
n 1 (cid:18) (cid:19)
X≥
j
2.2. The number of nodes labelled
j Z S S (t,u)
j j
Letusnowturnourattentiontoabivariate
ountingproblem. For ∈ ,let ≡
t
bethegeneratingfun
tionoflabelledtrees,
ountedbythenumberofedges(variable )and
j u S (t,1)=T(t) j
j
the number of nodes labelled (variable ). Clearly, for all . Moreover, an
S =S
j j
obvious symmetry entails that − .
j =0
Lemma 3. For 6 ,
S =1+t(S +S )S
j j 1 j+1 j
− (8)
j =0
while for ,
S =u+t(S +S )S =u+2tS S .
0 1 1 0 1 0
− (9)
S S (t,u)
j j
Proof. Observethat ≡ is alsothe generatingfun
tion of labelled treesrootedat
j 0
,
ounted by the number of edges and the number of nodes labelled . The de
omposition
of trees illustrated in Figure 4 then provides the lemma. The only di(cid:27)eren
e between the
j =0 j =0
ases and 6 lies in the generating fun
tion of the tree redu
ed to a single node.
S (t,u)
j
Again, the series turn out to be algebrai
, for reasons that
urrently remain
mysterious(fromthe
ombinatori
sviewpoint). They
anbeexpressedintermsoftheseries
T Z
and given by (5(cid:21)6).
j j Z
Proposition 4 (The number of nodes labelled ). For any ∈ , the generating
S S (t,u)
j j
fun
tion ≡ that
ounts labelled trees by the number of edges and the number of
j Q(T,u) 3
nodes labelled is algebrai
over of degree at most (and hen
e has degree at most
6 Q(t,u)
over ). More pre
isely,
(T S )2 2(1 T2)
0
− =1 − ,
(u 1)2 − 2+S S T (10)
0 0
− −
S Q(t,u,S ) j 0
j 0
and all the belong to . Moreover, for all ≥ ,
(1+µZj)(1+µZj+4)
S =T ,
j (1+µZj+1)(1+µZj+3) (11)
Z Z(t) µ µ(t,u) t
where ≡ is given by (5) and ≡ is the unique formal power series in
satisfying
(1+Z2)(1+µZ)(1+µZ2)(1+µZ3)
µ=(u 1) .
− (1+Z)(1+Z+Z2)(1 Z)3(1 µZ2) (12)
− −
µ(t,u) u µ(t,1) = 0 3
The series has polynomial
oe(cid:30)
ients in , and satis(cid:28)es . It has degree
Q(Z,u) 12 Q(t,u)
over and over .
µ Z
At some point, we will need a
losed form expression for in terms of . Here is one.
Proposition 5. Write
(u 1)Z(1+Z2)
v = − .
(1+Z)(1+Z+Z2)(1 Z)3
−
µ S
j
Then the algebrai
series involved in the expression (11) of , and de(cid:28)ned by (12), is
1 2
µ(t,u)= 1
Z2 1+v(1 Z)2/3+2/3 3+v2(1 Z)4cos(φ/3) − !
− −
p
8 MIREILLEBOUSQUET-MÉLOU
where 9v(1+4Z+Z2)+v3(1 Z)6
φ=arccos − − .
(3+v2(1 Z)4)3/2
(cid:18) − (cid:19)
S ,S ,S ,...
1 0 1 2
Proof of Propositions 4 and 5. First, observe that the family of series is
j > 0
ompletely determined by (8) (taken for ) and the se
ond part of (9). The fa
t that
µ Q(u)[[t]] j >0
for any series ∈ , the expression (11) satis(cid:28)es (8) for all is a straighforward
t T Z
veri(cid:28)
ation, on
e and have been expressed in terms of (see (5) and (6)). The form
S
j
of (11) is borrowed from [5℄. In order for (11) to be the
orre
t expression of , it remains
tosatisfythese
ondpartof (9). Thislast
onditionprovidesapolynomialequationrelating
µ T Z t u t T Z
, , , and . In this equation,repla
e and by their expressionsin terms of (given
µ 6 Q(T,u)
by (5(cid:21)6)). Thisgivesexa
tly (12). It
anbe easily
he
kedthat hasdegree over
12 Q(t,u)
and degree over .
S µ Z
0
Theequation(10)satis(cid:28)edby isobtainedby eliminating and (using (12)and (6))
S 6 Q(t,u)
0
from the expression (11) of . This equation gives an equation of degree over if
T
one eliminates thanks to (4).
j j 1
Nowtheequations(9),(8)and(4),
ombinedwithanindu
tionon ,implythatfor ≥ ,
S Q(T,u,S )
j 0
the series belongs to the (cid:28)eld , whi
h has just been proved to be an extension
Q(T,u) 3
of of degree . This
on
ludes the proof of Proposition 4.
µ
Let us (cid:28)nally prove Proposition 5. The equation (12) that de(cid:28)nes
an be rewritten
v (1+µZ)(1+µZ2)(1+µZ3)
µ= .
Z 1 µZ2
−
µ v Z
Hen
e is the unique formal powerseriesin (with rational
oe(cid:30)
ients in ) that satis(cid:28)es
0 v 0
the above equation and equals when is . It is not hard to
he
k that the
losed form
expression we give satis(cid:28)es these two
onditions.
Remarks
1. The produ
tform(11)ofProposition4re(cid:28)nestheprodu
tform(7)thatdealswithtrees
u=0 µ= 1 S (t,0)
j
with smalllabels. Indeed, when , Eq. (12) gives − , and the expressionof
T (t)
j 1
oin
ides, as it should, with the expression of − given by Proposition 2.
S
0
2. Thereexistsanalternativewaytoderiveanequationfor fromthesystemofLemma3.
Aswasobservedin[6, p. 645℄fortheproblemof
ountingtreeswithbounded labels,Eq.(8)
j 1
implies that for ≥ ,
I(S ,S )=I(S ,S )
j 1 j j j+1
−
I
where the (cid:16)invariant(cid:17) fun
tion is given by
I(x,y)=xy(1 tx)(1 ty)+txy x y.
− − − −
S T j t
j
But
onvergesto as goestoin(cid:28)nity,inthesetofformalpowerseriesin . Thisimplies
I(S ,S )=I(T,T).
0 1
S S T t
1 0
Eliminating between the above equation and (9) gives an equation between , and .
j
2.3. The number of nodes labelled or more
j Z R R (t,u)
j j
Letus (cid:28)nally study ourthird and last enumerationproblem. For ∈ , let ≡
t
bethegeneratingfun
tionoflabelledtrees,
ountedbythenumberofedges(variable )and
j u
the number of nodes labelled at least (variable ).
1
All the
al
ulations in this paper have been done using Maple. We do not re
ommend the reader to
he
kthembyhand.
LIMIT LAWS FOR EMBEDDED TREES 9
R ,R ,R ,...
0 1 2
Lemma 6. The set of series is
ompletely determined by the following equa-
j 1
tions: for ≥ ,
R =1+tR (R +R )
j j j 1 j+1
− (13)
and
R (t,u)=uR (tu,1/u).
0 1
(14)
j Z
More generally, for all ∈ , one has:
R (t,u)=uR (tu,1/u).
j j+1
− (15)
j Z R R (t,u)
j j
Proof. For all ∈ , the series ≡ is alsothe generatingfun
tion of treesrooted
j
at ,
ountedbytheirnumberofedgesandthenumberofnodeshavinganon-positivelabel.
j j 1
The equation satis(cid:28)ed by , for ≥ , follows on
e again from the de
omposition of trees
τ
illustrated in Figure 4. It remains to prove the symmetry relation (15). For any tree ,
n (τ) τ
0
let ≤ denote the number of nodes of having a non-positive label. We use similar
j
j,n
notations for the number of nodes having label at most , et
. Let T denote the set of
j n
trees rooted at and having edges. As observed above,
R j(t,u)= tn un≤0(τ) = tn un+1−n>0(τ),
−
nX≥0 τ∈XT−j,n nX≥0 τ∈XT−j,n
n n+1 1
be
auseatreewith edgeshasatotalof nodes. Atranslationofalllabelsby− gives
R j(t,u)=u (tu)n u−n≥0(τ),
−
nX≥0 τ∈TX−j−1,n
k k
while repla
ing ea
h label by − (cid:28)nally gives
R j(t,u)=u (tu)n u−n≤0(τ) =uRj+1(tu,1/u).
−
nX≥0 τ∈XTj+1,n
R T
j
Again, the series are algebrai
, and admit a
losed form expression in terms of and
Z
.
j j Z
Proposition 7 (The number of nodes labelled or more). Let ∈ . The generating
R (t,u) R
j j
fun
tion ≡ that
ounts labelled trees by the number of edges and the number of
j 2 Q(T(t),T(tu))
nodes labelled or more is algebrai
of degree at most over . Hen
e it has
8 Q(t,u) Q(T(t),T(tu))
degree at most over . More pre
isely, it belongs to the extension of
generated by
(T +T˜)2 4TT˜(T 1)(T˜ 1)
− − −
T T(t) T˜ T(tuq)
where ≡ and ≡ .
j 0
Moreover, for all ≥ ,
(1+νZj)(1+νZj+4)
R =T ,
j (1+νZj+1)(1+νZj+3) (16)
Z Z(t) ν ν(t,u) t
where ≡ is given by (5) and ≡ is a formal power series in , with polynomial
u 4 Q(u,Z) 16 Q(t,u)
oe(cid:30)
ients in , whi
h is algebrai
of degree over , and of degree over .
ν(t,1)=0
This series satis(cid:28)es . The (cid:28)rst terms in its expansion are:
ν(t,u)=(u 1) 1+2ut+ 7u+6u2 t2+ 32u+36u2+23u3 t3+O(t4) .
−
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1) ν (cid:17) ν
Before we prove this proposition, let us give something like a
losed form for . Sin
e
Q(u,Z) Z Q(t) ν
hasdegree4over ,and hasdegree4over , theseries isin theoryexpressible
in terms of radi
als... It turns that this expression is less terrible than one
ould fear.
10 MIREILLEBOUSQUET-MÉLOU
t
Proposition 8. De(cid:28)ne the following four formal power series in with polynomial
oe(cid:30)-
u
ients in : Z(1+Z2) 1 8tu
δ δ(t,u)=1 8(u 1) = − ,
≡ − − (1 Z)4 1 8t
− −
1 √δ 1− 11−88ttu
V V(t,u)= − = − ,
≡ 4 q4
4ZV2
∆ ∆(t,u)=(1 V)2 ,
≡ − − (1+Z)2
and 1 V √∆
P =(1+Z) − − .
2VZ
P 16 Q(t,u) 2 Q(V,Z)
Then has degree over , degree over , and satis(cid:28)es the following (cid:16)La-
grangian(cid:17) equation: V
P = (1+P)(1+ZP).
1+Z
ν R
j
Moreover, the algebrai
series involved in the expression (16) of is
P 1 P(1+Z) P2(1+Z+Z2)
ν = − − .
Z 1+Z+Z2+PZ(1+Z) P2Z2
−
Proof of Proposition 7. We have already
he
ked, in the proof of Proposition 4, that for
ν t j 0
any formal power series in , the series de(cid:28)ned by (16) for ≥ satisfy the re
urren
e
j 1 ν
relation (13) for ≥ . It reAmaints to prove that one
an
hoose u so as to satisfyA˜(14).
For anAy˜(tf,ourm)=alAp(otwue,r1/suer)ies in havinA˜˜g=raAtional
oe(cid:30)
ients in , wRe denote by the
j
series . Observe that . With this notation, if is of the generi
form (16), the relation (14) holds if and only if
T˜ (1+νZ)(1+νZ3)(1+ν˜Z˜)(1+ν˜Z˜5)
1+ν =u .
T (1+νZ4)(1+ν˜Z˜2)(1+ν˜Z˜4) (17)
R [u] u
m
Let denote the spa
e of polynomials in , with real
oe(cid:30)
ients, of degree at most
m R [u][[t]] t u
n
. Let denote the set of formal power series in with polynomial
oe(cid:30)
ients in
m n tm m
su
h that forall ≤ , the
oe(cid:30)
ient of has degreeat most . Observethat this set of
seriesin stable underthe usualoperationson series: sum, produ
t, and quasi-inverse. Write
ν = ν (u)tn n
n≥0 n ν (.uW) earegoingtoprove,byindu
tiononR, th[aut](17)determinesuniquely
n n+1
ea
hP
oe(cid:30)
ient , and that this
oe(cid:30)
ient belongs to .
ν u+O(t)
First, observe that for any formal power series , the right-hand side of (17) is .
ν (u)=u 1 m<n
0
This implies − . Now assume that our indu
tion hypothesis holds for all .
Z t νZ R [u][[t]]
n
Re
all that is a multiple of : this implies that belongs to . The indu
tion
tm uν˜ R [u] m < n
m+1
hypothesisZ˜al=soZim(tup)lie=sttuha+tOth(te2)
oe(cid:30)
ient of tin ubelongs to ,Rfor[ua]l[l[t]] .
n
Notethat ν˜Z˜ R [u][[its]]amultipleof and andalsobelongsto . This
n
implies that belongs to tooT. ,TT˜h,eZs,aZ˜me is true for all the other series o
urring
in the right-hand side of (17), namely . Given the
losure properties of the set
R [u][[t]] u
n
, we
on
lude thZat theZ˜right-hand side oft(17), divided by , belongs to thistnset.
Moreover, the fa
t that and are multiples of guarantees that the
oe(cid:30)
ient of in
ν (u) i < n tn
i
this series only involves the for . By extra
ting the
oe(cid:30)
ient of in (17), we
ν (u) uR [u] R [u]
n n n+1
on
lude that is uniquely determined and belongs to ⊂ .
ν
This
ompulet=es1the proofTo˜f=thTe existZe˜n=
eZand uniqueness of the series νs(att,is1f)y=ing0(17).
Also, setting (that is, and ) in this equation shows that .
t tu u 1/u
Let us now repla
e by and by in (17). This gives:
1 T (1+ν˜Z˜)(1+ν˜Z˜3)(1+νZ)(1+νZ5)
1+ν˜= .
u T˜ (1+ν˜Z˜4)(1+νZ2)(1+νZ4) (18)
