13 Pages
English

# The height and width of simple trees

-

13 Pages
English

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

Subjects

Informations

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from
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deep
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W
tree
has
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e
,
in
2])
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other
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h
d
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t
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ula
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W
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o
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[3,
=
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3]
exp(2
again.
)
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2
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+
f
2
l
(2
t
)
0
2
this
2
consequence
)
of
s
[3
c
,
quenc
Th.
for
3]
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=
)
w
1
to
e
b
t
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<
p
econd
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(
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+
out,
)
and
1
t
E
hat
W
the
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the
w

V
2
w
t
elcome
2
an
t
'elemen
s
tary'
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nd
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direct
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pro
Z
of.
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(
+
;
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ung
;
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2
)
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denote
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d
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onuen
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satisfy
o
o
h
pro

of
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t
pro
(
t
e;
suitably
=
the
where
w
is
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<
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+
a
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andom
s
oted
6
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=
w
0
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;
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1
on
;
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2
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a
normalized
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wnian
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xcursion
l
or
2),
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l
z
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yp
time
ergeometric
the
f
Bro
unction,
e
dened,
f
;
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the
tly
prole
1.1
pro
;
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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)