The height and width of simple trees

The height and width of simple trees

-

Documents
13 pages
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

Description

The height and width of simple trees P. Chassaing 1 , J.F. Marckert 1 , M. Yor 2 . The limit law of the couple height-width for simple trees can be seen as a consequence of deep results of Aldous, Drmota and Gittenberger, and Jeulin. We give here an elementary proof in the case of binary trees. 1 Introduction Let Z i (t) denote the number of nodes at distance i from the root of a rooted tree t. The prole of the tree t is the sequence (Z i (t)) i0 . The width w(t) and height h(t) of the tree t are dened by: w(t) = max i fZ i (t)g; h(t) = maxfijZ i (t) > 0g: Let T (n) B denote the set of binary trees with n leaves (2n 1 nodes), endowed with the uniform probability, and let H (n) B (resp. W (n) B ) be the restriction of h (resp. w) to T (n) B .

  • rst walk

  • galton watson tree

  • between height

  • standard normalized

  • brownian excursion

  • jeulin's description

  • normalized brownian

  • excursion - binary tree


Sujets

Informations

Publié par
Ajouté le 19 juin 2012
Nombre de lectures 9
Langue English
Signaler un abus

ert
nb
es
na
be
eg
be
from
The
)o
y:
the
of
with
des),
ed
be
the
of
y2
are
iv
la
V;
la
V;
Pr
)=
)e
be
the
the
on
[5,
9,
or
instance,
Then
H;
)=
eP
h
e
wnian
e
2
J.F.
VI,
s
o
a
1
consequence
tandard
of
al
deep
the
results
y:
of
+
Aldous,
:
5
].
0
BP
Gitten
width
,
robabilit
r
n
g
heigh
e
7
r
n
,
p
and

J
x
eulin.
distribution
W
other
1
)
i
Math
v
es
e
trees
here
eatoires
a
r
n
c
elemen
t
tary
,
pro
o
o
2
f
width
i
g
n
)
t
n
he
(
case
w
of
where:
binary
1
trees.
2
1
2
In
t
tro
et
duction
Theta
Let
in
Z
let
i
s
(
excursion
t
nstitut
)
V
denote
2
the
de
n
t
Cedex
a
Chassaing
with
r
0
aris
with
no
1
des
t
at
v
distance
progen
i
(see
.
27-28]).
P
la
ro
t
ot
[15
of
]
a
t
ro
13
5252
25]
tree
en
t
(
.
p
P
w
pr
+
ole
eles
of
)
the
n
tree
n
t
w
is
(
the
l
sequence
<
(
4
Z
2
i
2
(
2
t
Connections
))
een
i
V

2
0
and
.
or
The
are
width
1
w
,
(
(
t
0
)
1
and
Bro
height
Subsection
h
ematiques,
(
lie
t
54
trees
euvre
f
C
t
ersit
Marc
h
ab
e

tree
heigh
t
T
are
simple
dened
Galton-W
b

simple
ospring
w
istribution
(
o
t
2
)
p
=
y
7
/2,
i
onditioned
f
o
Z
a
i
e
(
otal
t
for
)
1
g
[1
;
pp.
h
Then,
(
limit
t
w
)
f
=
he
-
t
f
,
i
3
j
and
Z
f
i
he
(
[
t
,
)
,
>
t-width
0
heigh
g
b
:
H
Let
n
T
B
(
2
n
couple
)
!
B
!
denote
1
of
the
set
W
width
n
binary
B
trees
2
and
of
n
!
lea
!
v
1
es

Jussieu
a
n
V
1
x
no
X
t
<k
endo
+
w
d
heigh
k
with
x
place
limit
uniform
p
probabilit
k
y
x
,

4,
mo
-
The
H
w
(
.
n
of
)
on
B
hand,
(resp.
Bro
W
motion
(
Jacobi's
n
function
)
the
B
es
)
discussed
The
Y

20
1
F
ds
1
c
e
the
s
andom

ariables

k

a
denote
a
s
s
normalized

wnian
1
(see
o
3.1).
d
(
r
)
v
max
(

W

e
Z
6
0
restriction
e
o
s
f
;
h
0
(resp.
s
w
1
)
(
to
)
T
;
(
Lab
n
ratoire
)
e
B

.
I
One
E
5
Cartan
also
239
our
506
H
ando
(
l
n
Nancy
)
edex
B
Univ
and

W
aris
(
L
n
oratoire
)
Probabilit
B
as
d
hand,
one
(1.3)
(1
(1.2)
(1.1)
tree
atson
and
see
can
the
let
and
the
(2
max
max
oted
the
of
um
and
Drmota
M.la
la
tit
yi
69].
la
H;
be
width
tree
giv
and
23
of
be
erger
deserv
ed
ould
th
;

,b
(
;
)=
)(
+1
:::
H;
be
[11
help
of
form
or
;

(1
As
e,
Re
;R
and
Re
(1+2
)=
)(
of
Th.
3]
es
and
des
lo
al
,d
the
f
r
o
)
f
d
the

follo
f
wing
s
theorem:
(
Theorem


)

.
H

(
;
n
t
)
0
B
Aldous
p
c
2
=0
n
)
;
join
W
recen
(
p
n

)
2
B
tressed
p
a
2
)
n
s
!
+
s
t
w
2
!
that,
n
lab
!
e
+
+
1
k
(
!
iden

W
+
)
of
:
n
Note
Catherine
that
.
the
]),
ob
Theorem
vious

negativ

e
2
correlation

rst
=
t

w
rst
een
w
heigh
s
t
:
)
r
The
(
of
)
a
0
2
t
with
d
:
f
en
o
size
depth-rst
n
r
,
ith
s
j
reected
giv
in
z
the
X
dep

e
k
ndence
k
b
(
et
=
w

een
(
R
1).
1
la
0
of
ds
aim
e
v
(
b
s
The
)
the
2
agreemen
max
y:
0
obtains

wing
s
F

0
1
,
e
exp
(
2
s
2
).
in
Previous
2
results

(

,

H
;
]
:
e
onse
]
as
heigh
>
t
(
and
0
;a
s
=
>
simple
(
[0
t
b
2
e
s
longs
[9],
I
1
0
+
1
+
of
s
computer

science.
A
Surprisingly

,

Theorem
n
1.1
Theorem
do
,
es
v
not
rescaled,
s
alk
eem
of
t
ro
o
tree
b
no
e
v
stated
a
a
y:
n
to
ywhere,
;
though
is
Z
1
=
k
w
(
deduced
)
easily
z
from
(
deep
)
results
k
of
where
Aldous

o
k
n
(
)
er
hand
pap
(ab
this
o

ut
k
the
Z
con
t
tin
w
uum
(
random
W
tree
has
[1
e
,
in
2])
estigated
and
tly
o
y
n
Donati-Martin
the
]
other
With
h
d
and
the
o
t
f
ula
D
[22
rmota
she
&
the
Gitten
ollo
b
results:
W
b
[
,
12

],
[4
using
0
a
E
clev
W
x

idea
2
due
W
t

o
H
Aldous
W
[3,
=
(

3]
exp(2
again.
)
W
2
e
+
f
2
l
(2
t
)
0
2
this
2
consequence
)
of
s
[3
c
,
quenc
Th.
for
3]
(
=
)
w
1
to
e
b
t
e
<
p
econd
oin
(
ted
+
out,
)
and
1
t
E
hat
W
the
H
reader
the
w

V
2
w
t
elcome
2
an
t
'elemen
s
tary'
t
a
2
nd
Z
direct
1
pro
Z
of.
1
Let

(
+
;
2

ung
;
)
z
2
)
e
denote
)
the
d
c
:
onuen
First
satisfy
o
o
h
pro

of
[3
imple
C
t
pro
(
t
e;
suitably
=
the
where
w
is
e
<
e
+
a
1
andom
s
oted
6
eled
=
w
0
n
;
u
1
on
;
erges
2
oin
a
normalized
o
wnian
2
xcursion
l
or
2),
j
l
z
e
j
c
yp
time
ergeometric
the
f
Bro
unction,
e
dened,
f
;
ds:
dx
the
tly
prole
1.1
pro
;
5+
1.2
(see
The
hat
elt
Th.
er
one
can
it
foundations
the
to
trees
width
out
ab
[15
is
and
1.1
(1.4)