<s>
In	O
computing	O
,	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
a	O
term	O
for	O
the	O
value	O
of	O
a	O
tree	B-Application
data	I-Application
structure	I-Application
with	O
a	O
variable	O
and	O
unbounded	O
number	O
of	O
branches	O
per	O
node	O
.	O
</s>
<s>
The	O
term	O
is	O
mostly	O
used	O
in	O
the	O
functional	B-Language
programming	I-Language
community	O
,	O
e.g.	O
,	O
in	O
the	O
context	O
of	O
the	O
Bird	B-Application
–	I-Application
Meertens	I-Application
formalism	I-Application
.	O
</s>
<s>
Apart	O
from	O
the	O
multi-branching	O
property	O
,	O
the	O
most	O
essential	O
characteristic	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
is	O
the	O
coincidence	O
of	O
bisimilarity	B-Application
with	O
identity	B-Application
:	O
two	O
distinct	O
rose	B-Data_Structure
trees	I-Data_Structure
are	O
never	O
bisimilar	B-Application
.	O
</s>
<s>
The	O
name	O
"	O
rose	B-Data_Structure
tree	I-Data_Structure
"	O
was	O
coined	O
by	O
Lambert	O
Meertens	O
to	O
evoke	O
the	O
similarly	O
named	O
,	O
and	O
similarly	O
structured	O
,	O
common	O
rhododendron	O
.	O
</s>
<s>
We	O
shall	O
call	O
such	O
trees	O
rose	B-Data_Structure
trees	I-Data_Structure
,	O
a	O
literal	O
translation	O
of	O
rhododendron	O
(	O
Greek	O
=	O
rose	O
,	O
=	O
tree	O
)	O
,	O
because	O
of	O
resemblance	O
to	O
the	O
habitus	O
of	O
this	O
shrub	O
,	O
except	O
that	O
the	O
latter	O
does	O
not	O
grow	O
upside-down	O
on	O
the	O
Northern	O
hemisphere	O
.	O
</s>
<s>
Well-founded	B-Algorithm
rose	B-Data_Structure
trees	I-Data_Structure
can	O
be	O
defined	O
by	O
a	O
recursive	O
construction	O
of	O
entities	O
of	O
the	O
following	O
types	O
:	O
</s>
<s>
The	O
original	O
paper	O
only	O
considers	O
1+2b	O
(	O
"	O
sequence-forking	O
"	O
rose	B-Data_Structure
trees	I-Data_Structure
)	O
and	O
1+2a	O
(	O
"	O
set-forking	O
"	O
rose	B-Data_Structure
trees	I-Data_Structure
)	O
.	O
</s>
<s>
leaf	O
containing	O
a	O
value	O
,	O
or	O
a	O
node	O
that	O
can	O
have	O
an	O
arbitrary	O
list	O
of	O
subtrees	B-Application
.	O
</s>
<s>
The	O
most	O
common	O
definition	O
used	O
in	O
functional	B-Language
programming	I-Language
(	O
particularly	O
in	O
Haskell	B-Language
)	O
combines	O
3+2b	O
:	O
</s>
<s>
An	O
element	O
of	O
Rose	O
α	O
consists	O
of	O
a	O
labelled	O
node	O
together	O
with	O
a	O
list	O
of	O
subtrees	B-Application
.	O
</s>
<s>
That	O
is	O
,	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
a	O
pairing	O
entity	O
(	O
type	O
3	O
)	O
whose	O
branching	O
entity	O
is	O
a	O
sequence	O
(	O
thus	O
of	O
type	O
2b	O
)	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
.	O
</s>
<s>
General	O
rose	B-Data_Structure
trees	I-Data_Structure
can	O
be	O
defined	O
via	O
bisimilarity	B-Application
of	O
accessible	O
pointed	O
multidigraphs	O
with	O
appropriate	O
labelling	O
of	O
nodes	O
and	O
arrows	O
.	O
</s>
<s>
Two	O
apqs	O
and	O
are	O
said	O
to	O
be	O
bisimilar	B-Application
if	O
there	O
exists	O
a	O
bisimilarity	B-Application
relation	O
for	O
them	O
.	O
</s>
<s>
A	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
then	O
some	O
fixed	O
representation	O
of	O
the	O
class	O
of	O
apqs	O
that	O
are	O
bisimilar	B-Application
to	O
some	O
given	O
apq	O
.	O
</s>
<s>
If	O
the	O
root	B-Application
node	I-Application
of	O
is	O
of	O
type	O
(	O
1	O
)	O
then	O
}	O
,	O
thus	O
can	O
be	O
represented	O
by	O
this	O
root	B-Application
node	I-Application
.	O
</s>
<s>
As	O
a	O
result	O
of	O
the	O
above	O
set-theoretic	O
construction	O
,	O
the	O
class	O
of	O
all	O
rose	B-Data_Structure
trees	I-Data_Structure
is	O
defined	O
,	O
depending	O
on	O
the	O
sets	O
(	O
ground	O
values	O
)	O
,	O
(	O
arrow	O
names	O
)	O
and	O
(	O
node	O
labels	O
)	O
as	O
the	O
definitory	O
constituents	O
.	O
</s>
<s>
For	O
every	O
element	O
of	O
there	O
is	O
an	O
induced	O
apq	O
such	O
that	O
is	O
the	O
root	B-Application
node	I-Application
of	O
and	O
the	O
respective	O
sets	O
and	O
of	O
nodes	O
and	O
arrows	O
of	O
are	O
formed	O
by	O
those	O
elements	O
of	O
and	O
that	O
are	O
accessible	O
via	O
a	O
path	O
of	O
arrows	O
starting	O
at	O
.	O
</s>
<s>
The	O
induced	O
apq	O
is	O
bisimilar	B-Application
to	O
apqs	O
used	O
for	O
the	O
construction	O
of	O
.	O
</s>
<s>
Rose	B-Data_Structure
trees	I-Data_Structure
that	O
do	O
not	O
contain	O
set-branching	O
nodes	O
(	O
type	O
2a	O
)	O
can	O
be	O
represented	O
by	O
pathname	O
maps	O
.	O
</s>
<s>
The	O
target	O
of	O
the	O
empty	O
path	O
is	O
the	O
root	B-Application
node	I-Application
.	O
</s>
<s>
Given	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
that	O
does	O
not	O
contain	O
set-branching	O
nodes	O
,	O
the	O
pathname	O
map	O
of	O
is	O
a	O
map	O
that	O
assigns	O
each	O
resolvable	O
pathname	O
its	O
value	O
according	O
to	O
the	O
following	O
general	O
scheme	O
:	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
different	O
rose	B-Data_Structure
trees	I-Data_Structure
have	O
different	O
pathname	O
maps	O
.	O
</s>
<s>
For	O
"	O
homogeneous	O
"	O
rose	B-Data_Structure
trees	I-Data_Structure
there	O
is	O
no	O
need	O
for	O
type	O
tagging	O
,	O
and	O
their	O
pathname	O
map	O
can	O
be	O
defined	O
as	O
summarized	O
below	O
:	O
</s>
<s>
In	O
particular	O
,	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
in	O
the	O
most	O
common	O
"	O
Haskell	B-Language
"	O
sense	O
is	O
just	O
a	O
map	O
from	O
a	O
non-empty	O
prefix-closed	O
and	O
left-sibling-closed	O
set	O
of	O
finite	O
sequences	O
of	O
natural	O
numbers	O
to	O
a	O
set	O
.	O
</s>
<s>
Such	O
a	O
definition	O
is	O
mostly	O
used	O
outside	O
the	O
branch	O
of	O
functional	B-Language
programming	I-Language
,	O
see	O
Tree	O
(	O
automata	O
theory	O
)	O
.	O
</s>
<s>
Typically	O
,	O
documents	O
that	O
use	O
this	O
definition	O
do	O
not	O
mention	O
the	O
term	O
"	O
rose	B-Data_Structure
tree	I-Data_Structure
"	O
at	O
all	O
.	O
</s>
<s>
The	O
diagrams	O
below	O
show	O
two	O
examples	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
together	O
with	O
the	O
correspondent	O
Haskell	B-Language
code	O
.	O
</s>
<s>
In	O
both	O
cases	O
,	O
the	O
Data.Tree	O
module	O
is	O
used	O
as	O
it	O
is	O
provided	O
by	O
the	O
Haskell	B-Language
containers	O
package	O
.	O
</s>
<s>
The	O
module	O
introduces	O
rose	B-Data_Structure
trees	I-Data_Structure
as	O
pairing	O
entities	O
by	O
the	O
following	O
definition	O
:	O
</s>
<s>
Both	O
examples	O
are	O
contrived	O
so	O
as	O
to	O
demonstrate	O
the	O
concept	O
of	O
"	O
sharing	O
of	O
substructures	O
"	O
which	O
is	O
a	O
distinguished	O
feature	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
.	O
</s>
<s>
(	O
so	O
that	O
the	O
labels	O
'	O
a	O
 '	O
,	O
'	O
b	O
 '	O
,	O
'	O
c	O
 '	O
or	O
'	O
d	O
 '	O
uniquely	O
identify	O
a	O
subtree	B-Application
/	O
node	O
)	O
which	O
does	O
not	O
need	O
to	O
be	O
satisfied	O
in	O
general	O
.	O
</s>
<s>
In	O
each	O
of	O
the	O
examples	O
,	O
the	O
rose	B-Data_Structure
tree	I-Data_Structure
in	O
question	O
is	O
labelled	O
by	O
'	O
a	O
 '	O
and	O
equals	O
the	O
value	O
of	O
the	O
a	O
variable	O
in	O
the	O
code	O
.	O
</s>
<s>
The	O
first	O
example	O
presents	O
a	O
well-founded	B-Algorithm
rose	B-Data_Structure
tree	I-Data_Structure
a	O
obtained	O
by	O
an	O
incremental	O
construction	O
.	O
</s>
<s>
The	O
rose	B-Data_Structure
tree	I-Data_Structure
can	O
be	O
represented	O
by	O
the	O
pathname	O
map	O
shown	O
on	O
the	O
left	O
.	O
</s>
<s>
The	O
second	O
example	O
presents	O
a	O
non-well-founded	O
rose	B-Data_Structure
tree	I-Data_Structure
a	O
built	O
by	O
a	O
breadth-first	O
constructor	O
unfoldTree	O
.	O
</s>
<s>
The	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
a	O
Moore	B-General_Concept
machine	I-General_Concept
,	O
see	O
notes	O
above	O
.	O
</s>
<s>
The	O
general	O
definition	O
provides	O
a	O
connection	O
to	O
tree	B-Application
data	I-Application
structures	I-Application
:	O
</s>
<s>
Rose	B-Data_Structure
trees	I-Data_Structure
are	O
tree	O
structures	O
modulo	O
bisimilarity	B-Application
.	O
</s>
<s>
Every	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
bisimilar	B-Application
to	O
such	O
a	O
tree	O
structure	O
(	O
since	O
every	O
apq	O
is	O
bisimilar	B-Application
to	O
its	O
unfolding	O
)	O
and	O
every	O
such	O
tree	O
structure	O
is	O
bisimilar	B-Application
to	O
exactly	O
one	O
rose	B-Data_Structure
tree	I-Data_Structure
which	O
can	O
therefore	O
be	O
regarded	O
as	O
the	O
value	O
of	O
the	O
tree	O
structure	O
.	O
</s>
<s>
In	O
the	O
lower	O
part	O
,	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
shown	O
that	O
is	O
the	O
value	O
of	O
.	O
</s>
<s>
There	O
is	O
an	O
induced	O
subtree-to-subvalue	O
mapping	O
which	O
is	O
partially	O
displayed	O
by	O
blue	O
arrows	O
.	O
</s>
<s>
Observe	O
that	O
the	O
mapping	O
is	O
many-to-one	O
:	O
distinct	O
tree	B-Application
data	I-Application
structures	I-Application
can	O
have	O
the	O
same	O
value	O
.	O
</s>
<s>
As	O
a	O
particular	O
consequence	O
,	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
in	O
general	O
is	O
not	O
a	O
tree	O
in	O
terms	O
of	O
"	O
subvalue	O
"	O
relationship	O
between	O
its	O
subvalues	O
,	O
see	O
#Terminological_controversy	O
.	O
</s>
<s>
The	O
value	O
mapping	O
described	O
above	O
can	O
be	O
used	O
to	O
clarify	O
the	O
difference	O
between	O
the	O
terms	O
"	O
tree	B-Application
data	I-Application
structure	I-Application
"	O
and	O
"	O
tree	O
data	O
type	O
"	O
:	O
</s>
<s>
A	O
tree	O
data	O
type	O
is	O
a	O
set	O
of	O
values	O
of	O
tree	B-Application
data	I-Application
structures	I-Application
.	O
</s>
<s>
This	O
becomes	O
apparent	O
when	O
one	O
compares	O
a	O
single	O
tree	O
data	O
type	O
with	O
a	O
single	O
tree	B-Application
data	I-Application
structure	I-Application
.	O
</s>
<s>
A	O
single	O
tree	O
data	O
type	O
contains	O
(	O
infinitely	O
)	O
many	O
values	O
each	O
of	O
which	O
is	O
represented	O
by	O
(	O
infinitely	O
)	O
many	O
tree	B-Application
data	I-Application
structures	I-Application
.	O
</s>
<s>
For	O
example	O
,	O
given	O
a	O
set	O
}	O
of	O
labels	O
,	O
the	O
set	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
in	O
the	O
Haskell	B-Language
sense	O
(	O
3b	O
)	O
with	O
labels	O
taken	O
from	O
is	O
a	O
single	O
tree	O
data	O
type	O
.	O
</s>
<s>
All	O
the	O
above	O
examples	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
belong	O
to	O
this	O
data	O
type	O
.	O
</s>
<s>
As	O
it	O
can	O
be	O
observed	O
in	O
the	O
above	O
text	O
and	O
diagrams	O
,	O
the	O
term	O
"	O
rose	B-Data_Structure
tree	I-Data_Structure
"	O
is	O
controversial	O
.	O
</s>
<s>
A	O
rose	B-Data_Structure
tree	I-Data_Structure
 [ ... ] 	O
is	O
either	O
a	O
leaf	O
 [ ... ] 	O
or	O
a	O
node	O
 [ ... ] 	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
definition	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
in	O
the	O
most	O
common	O
Haskell	B-Language
sense	O
suggests	O
that	O
(	O
within	O
the	O
context	O
of	O
discourse	O
)	O
"	O
node	O
"	O
and	O
"	O
tree	O
"	O
are	O
synonyms	O
.	O
</s>
<s>
Does	O
it	O
mean	O
that	O
every	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
coincident	O
with	O
its	O
root	B-Application
node	I-Application
?	O
</s>
<s>
If	O
so	O
,	O
is	O
such	O
a	O
property	O
considered	O
specific	O
to	O
rose	B-Data_Structure
trees	I-Data_Structure
or	O
does	O
it	O
also	O
apply	O
to	O
other	O
trees	O
?	O
</s>
<s>
one	O
can	O
conclude	O
that	O
rose	B-Data_Structure
trees	I-Data_Structure
in	O
general	O
are	O
not	O
trees	O
in	O
usual	O
meaning	O
known	O
from	O
computer	O
science	O
.	O
</s>
<s>
There	O
is	O
at	O
least	O
one	O
adoption	O
of	O
the	O
term	O
"	O
rose	B-Data_Structure
tree	I-Data_Structure
"	O
in	O
computer	O
science	O
in	O
which	O
"	O
sharing	O
of	O
substructures	O
"	O
is	O
precluded	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
Bayesian	O
rose	B-Data_Structure
tree	I-Data_Structure
is	O
based	O
on	O
the	O
following	O
definition	O
of	O
rose	B-Data_Structure
trees	I-Data_Structure
:	O
</s>
<s>
is	O
a	O
rose	B-Data_Structure
tree	I-Data_Structure
if	O
either	O
}	O
for	O
some	O
data	O
point	O
or	O
}	O
where	O
'	O
s	O
are	O
rose	B-Data_Structure
trees	I-Data_Structure
over	O
disjoint	O
sets	O
of	O
data	O
points	O
.	O
</s>
