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ts.
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generation
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or
haeer
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1997;
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et
b
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1994;
2002;
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haeer
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tation
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ting
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oted
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its
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b
o
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v
and
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maps
to
and
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hellen
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erg
1990;
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di
h
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et
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al.
ws.
1999;
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text
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mathematical
t's
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references
ordering
[Am
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t
al.
1996;
P
Ch
and
uang
haeer
et
and
al.
arious
1998;
yp
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hon
dom
et
graphs
al.
2003;
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graph
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w
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Journal
b
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y
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of
dirsky
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ha
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e
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h
so-called
literature
e
particular
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the
d
binatorial
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of
uis
meshes.
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algorithms
1978;
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triangular
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et
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and
1994;
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man
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the
if
resulting
edge
parametric
the
der
CM
is
Name,
more
ol.
No.
t
Mon
in
20YY.Dissections
and
trees
manner
j
d
B
5
trees
ot-no
planar
on
graphs
ed
t
maps
th
deriv
of
ed
of
maps
a
later
quadrangulations
n
deriv
to
ed
tree
maps
hoice
with
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orien
e
tation
ot-edge
dissections
of
r
the
wing
hexagon
no
dissections
t
of
trees
the
w
hexagon
of
with
+
orien
ro
tation
Catalan
binary
trees
in
paren
The
thesis
de.
white-r
de
Journal
iter
v
ative
a
algorithm
e
tr
stem.
ansp
binary
osition
op
a
er
.
ations
tree,
op
a
ening
ro
ro
e
son
r
ery
eje
of
Whitney
n
folklor
ectiv
e
0
Fig.
oted
1.
ha
Relations
es,
b
on
et
w
w
ted
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ers:
in
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v
olv
greedily
ed
white
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v
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ed
o
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orien
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ted
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ro
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ro
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th
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ys
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oted
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on
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tree
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do
origin
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of
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ro
de)
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is
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r
v
o
usual
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oted
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enien
F
and
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edges
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said
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e
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e
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y
on
).
whether
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e
y
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um
to
0
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1
outer
n
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not.
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sa
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map
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onne
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rst
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y
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r
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o
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With
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ro
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up
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ectiv
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ely
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ro
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ariation
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ro
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e
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ely
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y
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ro
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es
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y
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o
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ote
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o
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es
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are
on
plane
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whose
ot
no
CM
des
Name,
ha
ol.
v
No.
e
Mon
degree
20YY.6
?ric
F
e
OF
usy
D
et
al.
n
no
disse
de.
sa
The
and
sets
of
of
adaptation
blac
resp
k-ro
dissections
oted
are
(resp.
v
white-ro
k
oted)
ro
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trees
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with
t
i
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k
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and
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.
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of
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h
b
b
y
k
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y
on
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Æ
and
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and
);
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set
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with
denoted
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n
k
dissections
no
degree,
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and
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edge
white
a
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ro
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v
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een
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angular
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h
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en
us
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e
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as
th
a
ely)
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t
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ro
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ectiv
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edge
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as
maps
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greedily
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to
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e
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e
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h
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y
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e
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is
MAPS
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to
hereafter
asso
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hellen
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ro
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oted,
ectiv
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e
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angulation
The
is
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oted
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ely
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n
e
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n
of
As
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es
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y
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v
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e
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ar
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ro
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b
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e,
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edges
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denote
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b
BETWEEN
y
AMILIES
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This
n
the
folklore
set
b
of
w
ro
oted
and
maps,
quadrangulations
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mapping,
n
[Mullin
Sc
b
the
℄
outer
its
one.
to
Euler's
relation
3.1
ensures
maps
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rst
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ho
v
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e
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orks.
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endo
denote
ed
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its
y
ertex
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let
n
b
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the
D
oted
0
obtained
n
CM
)
Name,
the
ol.
set
No.
of
Mon
(ro
20YY.Dissections
and
trees
and
Pr
e
D
7
(a)
yields
A
t
quadrangulation
irr
(b)
th
with
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its
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k
dissection
diagonals
y
(c)
omplete
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outer-triangular
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map.
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etwe
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et
angular
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to
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ro
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oted
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ro
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oted
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h
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triangular
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ery
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angular
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ro
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see
linking
Figure
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3.
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.
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ery
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8.
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black
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es
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v
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es.
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ro
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ot
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3.1,
the
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edge
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b
follo
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ws
CM
the
Name,
ro
ol.
ot
No.
of
Mon
M
20YY.8
?ric
F
dge
dual
dissections.
usy
the
et
al.
link
(a)
ed
A
example,
dissection,
dual
(b)
ertex
blac
of
k
primal
diagonals,
f
(c)
v
the
M
3(d).
map,
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of
the
ed
deriv
one
ed
D
map.
Fig.
3.
o
The
h
angular
ertex
mapping
eeping
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and
b
h
order:
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of
a
vertic
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primal
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dissection
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to
V,
an
v
outer-triangular
maps
map
ed
(c).
as
The
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o
deriv
f
ed
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map
y
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w
sho
w
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dge-vertex
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obtained
3.3
dual
Derived
three
maps
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In
regularit
its
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an
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outer
for
map
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e
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en
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e
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ery
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this
t
k
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t
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with
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orien
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e
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y
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sup
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ed
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to
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to
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ed
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o
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of
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ertex
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for
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or
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e
ed
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Figure
due
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to
wn
the
Figure
in
Blac
tersection
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e
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are
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onds
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when
ed
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Clearly
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outer
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y
a
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dissection
ertex,
its
or
map
to
v
an
the
half-edge
deriv
of
map.
M
CM
Name,
when
ol.
it
No.
Mon
an
20YY.Dissections
and
trees
e
e
ounded
the
9
to
(a)
A
n
binary
2
tree,
length
(b)
a
Journal
lo
to
(c)
um
and
inner
the
Let
partial
als
the
Fig.
the
4.
s
The
paren
partial
the
4.
to
BIJECTION
binary
BETWEEN
outer
BINARY
b
TREES
tire
AND
the
IRREDUCIBLE
v
DISSECTIONS
outer
4.1
Closure
er
mapping:
um
from
um
trees
r
to
.
dissections
t
Lo
of
for
and
is
pa
th
rtial
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in
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w
en
obtain
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an
with
edge
en
Observ
tire
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edges
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and
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and
er
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tree),
w
T
e
en
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in
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lo
terv
2,
al
b
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e
in
op
b
eration,
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(that
h
no
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based
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2
a
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y
kwise
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illustrates
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r
around
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the
Figure
map:
V,
this
of
w
w
alk
of
alongside
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the
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us
oundary
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of
eration
the
dissection
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with
map
outer
visits
dge
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b
en
of
stems
outer
and
that
en
T
tire
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2
or
n
more
tire
precisely
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,
b
a
n
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y
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um
ha
outer
ving
Hence,
the
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er
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there
righ
6
t-hand
half-edges.
side.
stems
When
als
a
on
stem
of
is
these
immediately
ha
follo
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ed
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in
p
this
b
w
um
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als
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er
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edges,
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lo
er
w
stems).
in
s
the
b
r
of
=
an
opp
osite
unmatc
half-edge
half-edges
for
v
this
hexagon
stem,
w
whic
to
h
hexagon)
is
quadrangular
Figure
hed
to
farthest
2
endp
2)
oin
particular
t
en
of
A
the
V
third
N,
en
hings
tire
the
edge:
thesis
this
ord.
amoun
example
ts
partial
to
is
wn
the
Figure
stem
Complete
in
Let
to
no
an
en
partial
tire
op
edge,
to
so
a
as
of
to
hexagon
quadrangular
or
An
entir
half-e
a
is
quadrangular
half-edge
elonging
This
an
op
tire
eration
and
is
t
illustrated
the
in
Figure
e
4(b).
a
Giv
tree
en
with
a
no
binary
has
tree
+
T
stems
,
2
the
2
lo
en
half-edges.
h
b
e
y
p
the
erformed
um
greedily
er
un
stems
til
b
no
2
more
n
lo
b
of
en
is
half-edges.
p
if
ossible.
denotes
n
h
b
lo
of
hed)
in
partial
a
of
new
,
en
are
tire
k
edge,
outer
ma
tire
yb
Moreo
e
er,
making
delimit
a
terv
new
of
lo
half-edges
the
tour
p
the
ossible.
It
in
is
als
easy
v
to
length
see
most
that
otherwise
the
lo
nal
map,
ould
e
the
ossible.
p
r
artial
e
n
e
b
of
of
T
h
,
terv
do
of
es
1
not
s
dep
e
end
n
on
b
the
of
order
h
of
terv
the
of
lo
0
is,
n
Indeed,
b
a
of
des
paren
t
thesis
t
w
o
ord
hed
is
Then
asso
and
are
to
related
the
y
relation
+
s
kwise
6
b
The
oundary
omplete
of
e
the
in
tree,
all
with
hed
an
with
op
ening
to
paren
thesis
the
of
in
w
unique
eigh
a
t
(up
3
rotation
for
the
a
that
stem
only
and
b
a
5(a)
paren
the
thesis
for
the
a
(
side
=
of
;
en
=
tire
,
edge;
a
then
example
the
giv
future
in
lo
5(b).
CM
Name,
ol.
ond
No.
to
Mon
matc
20YY.10
?ric
F
the
if
outwar
usy
that
et
V,
al.
v
its
oriente
a
to
w
half-edges
see
inner
orien
is
dissection
(a)
with
edges
orien
when
of
r
half-edges,
=
n
2
tri-orientation
and
that
s
of
=
to
2
ely
.
inner
(b)
An
Case
ha
of
other
the
w
binary
A
tree
are
of
righ
Figure
of
4(a).
An
Fig.
d
5.
ard
The
is
a
orien
Lemma
e
4.1.
naturally
The
of
w
e
tree
of
its
a
de
binary
an
tr
an
e
half-
e
h
is
v
an
3,
irr
half-edges
e
b
ard,
disse
to
t
of
the
w
hexagon.
ard),
Pr
oth
oof.
b
Assume
with
that
there
exists
or
a
of
separating
Journal
th
C
tation,
in
orien
the
is
e
of
it
T
to
.
origin
Let
if
m
ted
origin.
1
is
b
with
e
of
the
outde
n
of
um
v
b
as
er
b
of
v
ted
The
in
a
the
dened
in
tation
terior
of
y
C
outdegree
.
6(a)
Then
A
there
dissection
are
tation
2
half-edges
m
b
edges
edges)
in
outer
the
in
resp
terior
0
of
C
w
a
to
Euler's
b
relation.
in
Let
Figure
v
is
b
e
e
if
a
o
v
e
ertex
is,
of
ted
T
and
that
out
b
bi-oriente
elongs
are
to
ted
the
Let
in
an
terior
w
of
tri-orien
C
after
D
the
edges
Consider
bi-orien
the
orien
orien
in
tation
on
of
A
edges
V
of
N,
T
notion
a
orien
w
where
a
are
y
ted).
from
half-edge
v
said
(only
b
for
inwar
the
if
sak
is
e
ted
of
w
this
its
pro
and
of
d
).
it
Then
orien
no
out
des
its
of
If
T
map
ha
endo
v
ed
e
an
outdegree
tation
2,
its
the
v
gr
,
e
whic
a
h
ertex
has
is
outdegree
dened
3.
the
This
um
orien
er
tation
its
naturally
t
orien
an
out
orien
ard.
tation
(unique)
of
of
edges
binary
of
is
the
as
orien
with
of
the
half-edges
same
h
prop
an
ert
no
y
has
(except
3,
that
Figure
v
for
example.
of
tri-orientation
the
a
hexagon
is
ha
orien
v
of
e
inner
outdegree
(i.e.,
0).
edges
Hence
elonging
there
inner
are
at
that
least
and
2
v
m
ha
+
e
1
ectiv
edges
outdegree
in
and
the
and
in
h
terior
t
of
o
C
of
,
same
a
edge
not
tradiction.
oth
4.2
e
T
ted
ri-o
w
rientations
see
and
6(b).
op
edge
ening
said
T
b
ri-o
simply
rientations.
d
In
its
order
w
to
half-edges
dene
v
the
same
mapping
(that
in
one
v
orien
erse
in
to
ard
the
the
one
w
w
e
and
need
d
a
they
b
b
etter
orien
description
out
of
ard.
the
D
e
on
endo
the
ed
a
map
tation.
b
y
the
of
original
is
tree.
simple
Let
C
us
of
that
orien
either
tations
ted
of
simply
the
ted
half-e
the
dges
terior
of
C
a
their
map
t.
(in
CM
Name,
trast
ol.
to
No.
the
Mon
usual
20YY.
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