<s>
Space-filling	B-Algorithm
trees	I-Algorithm
are	O
geometric	O
constructions	O
that	O
are	O
analogous	O
to	O
space-filling	B-Algorithm
curves	I-Algorithm
,	O
but	O
have	O
a	O
branching	O
,	O
tree-like	O
structure	O
and	O
are	O
rooted	O
.	O
</s>
<s>
A	O
space-filling	B-Algorithm
tree	I-Algorithm
is	O
defined	O
by	O
an	O
incremental	O
process	O
that	O
results	O
in	O
a	O
tree	O
for	O
which	O
every	O
point	O
in	O
the	O
space	O
has	O
a	O
finite-length	O
path	O
that	O
converges	B-Algorithm
to	O
it	O
.	O
</s>
<s>
In	O
contrast	O
to	O
space-filling	B-Algorithm
curves	I-Algorithm
,	O
individual	O
paths	O
in	O
the	O
tree	O
are	O
short	O
,	O
allowing	O
any	O
part	O
of	O
the	O
space	O
to	O
be	O
quickly	O
reached	O
from	O
the	O
root	O
.	O
</s>
<s>
The	O
simplest	O
examples	O
of	O
space-filling	B-Algorithm
trees	I-Algorithm
have	O
a	O
regular	O
,	O
self-similar	O
,	O
fractal	B-Application
structure	O
,	O
but	O
can	O
be	O
generalized	O
to	O
non-regular	O
and	O
even	O
randomized/Monte	O
-Carlo	O
variants	O
(	O
see	O
Rapidly	B-General_Concept
exploring	I-General_Concept
random	I-General_Concept
tree	I-General_Concept
)	O
.	O
</s>
<s>
Space-filling	B-Algorithm
trees	I-Algorithm
have	O
interesting	O
parallels	O
in	O
nature	O
,	O
including	O
fluid	O
distribution	O
systems	O
,	O
vascular	O
networks	O
,	O
and	O
fractal	B-Application
plant	O
growth	O
,	O
and	O
many	O
interesting	O
connections	O
to	O
L-systems	B-Language
in	O
computer	O
science	O
.	O
</s>
<s>
A	O
space-filling	B-Algorithm
tree	I-Algorithm
is	O
defined	O
by	O
an	O
iterative	O
process	O
whereby	O
a	O
single	O
point	O
in	O
a	O
continuous	O
space	O
is	O
connected	O
via	O
a	O
continuous	O
path	O
to	O
every	O
other	O
point	O
in	O
the	O
space	O
by	O
a	O
path	O
of	O
finite	O
length	O
,	O
and	O
for	O
every	O
point	O
in	O
the	O
space	O
,	O
there	O
is	O
at	O
least	O
one	O
path	O
that	O
converges	B-Algorithm
to	O
it	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
"	O
space-filling	B-Algorithm
tree	I-Algorithm
"	O
in	O
this	O
sense	O
was	O
described	O
in	O
Chapter	O
15	O
of	O
Mandelbrot	O
's	O
influential	O
book	O
The	O
Fractal	B-Application
Geometry	O
of	O
Nature	O
(	O
1982	O
)	O
.	O
</s>
<s>
The	O
concept	O
was	O
made	O
more	O
rigorous	O
and	O
given	O
the	O
name	O
"	O
space-filling	B-Algorithm
tree	I-Algorithm
"	O
in	O
a	O
2009	O
tech	O
report	O
that	O
defines	O
"	O
space-filling	O
"	O
and	O
"	O
tree	O
"	O
differently	O
than	O
their	O
traditional	O
definitions	O
in	O
mathematics	O
.	O
</s>
<s>
As	O
explained	O
in	O
the	O
space-filling	B-Algorithm
curve	I-Algorithm
article	O
,	O
in	O
1890	O
,	O
Peano	O
found	O
the	O
first	O
space-filling	B-Algorithm
curve	I-Algorithm
,	O
and	O
by	O
Jordan	O
's	O
1887	O
definition	O
,	O
which	O
is	O
now	O
standard	O
,	O
a	O
curve	O
is	O
a	O
single	O
function	O
,	O
not	O
a	O
sequence	O
of	O
functions	O
.	O
</s>
<s>
The	O
curve	O
is	O
"	O
space	O
filling	O
"	O
because	O
it	O
is	O
"	O
a	O
curve	O
whose	O
range	O
contains	O
the	O
entire	O
2-dimensional	O
unit	O
square	O
"	O
(	O
as	O
explained	O
in	O
the	O
first	O
sentence	O
of	O
space-filling	B-Algorithm
curve	I-Algorithm
)	O
.	O
</s>
<s>
In	O
contrast	O
,	O
a	O
space-filling	B-Algorithm
tree	I-Algorithm
,	O
as	O
defined	O
in	O
the	O
tech	O
report	O
,	O
is	O
not	O
a	O
single	O
tree	O
.	O
</s>
<s>
The	O
paper	O
says	O
"	O
A	O
space-filling	B-Algorithm
tree	I-Algorithm
is	O
actually	O
defined	O
as	O
an	O
infinite	O
sequence	O
of	O
trees	O
"	O
.	O
</s>
<s>
It	O
defines	O
as	O
a	O
"	O
sequence	O
of	O
trees	O
"	O
,	O
then	O
states	O
"	O
is	O
a	O
space-filling	B-Algorithm
tree	I-Algorithm
"	O
.	O
</s>
<s>
It	O
states	O
"	O
A	O
tree	O
sequence	O
T	O
is	O
called	O
'	O
space	O
filling	O
 '	O
in	O
a	O
space	O
X	O
if	O
for	O
every	O
xX	O
,	O
there	O
exists	O
a	O
path	O
in	O
the	O
tree	O
that	O
starts	O
at	O
the	O
root	O
and	O
converges	B-Algorithm
tox.	O
"	O
.	O
</s>
<s>
The	O
simplest	O
example	O
of	O
a	O
space-filling	B-Algorithm
tree	I-Algorithm
is	O
one	O
that	O
fills	O
a	O
square	O
planar	O
region	O
.	O
</s>
<s>
Space-filling	B-Algorithm
trees	I-Algorithm
can	O
also	O
be	O
defined	O
for	O
a	O
variety	O
of	O
other	O
shapes	O
and	O
volumes	O
.	O
</s>
<s>
The	O
first	O
six	O
iterations	O
of	O
the	O
triangle	O
space-filling	B-Algorithm
tree	I-Algorithm
are	O
illustrated	O
below	O
:	O
</s>
<s>
Space-filling	B-Algorithm
trees	I-Algorithm
can	O
also	O
be	O
constructed	O
in	O
higher	O
dimensions	O
.	O
</s>
<s>
The	O
simplest	O
examples	O
are	O
cubes	B-Application
in	O
and	O
hypercubes	B-Operating_System
in	O
.	O
</s>
<s>
A	O
similar	O
sequence	O
of	O
iterations	O
used	O
for	O
the	O
square	O
space-filling	B-Algorithm
tree	I-Algorithm
can	O
be	O
used	O
for	O
hypercubes	B-Operating_System
.	O
</s>
<s>
The	O
third	O
iteration	O
of	O
such	O
a	O
space-filling	B-Algorithm
tree	I-Algorithm
in	O
is	O
illustrated	O
below	O
:	O
</s>
