<s>
In	O
fractal	B-Application
geometry	O
,	O
the	O
H	B-Algorithm
tree	I-Algorithm
is	O
a	O
fractal	B-Application
tree	O
structure	O
constructed	O
from	O
perpendicular	O
line	O
segments	O
,	O
each	O
smaller	O
by	O
a	O
factor	O
of	O
the	O
square	O
root	O
of	O
2	O
from	O
the	O
next	O
larger	O
adjacent	O
segment	O
.	O
</s>
<s>
An	O
H	B-Algorithm
tree	I-Algorithm
can	O
be	O
constructed	O
by	O
starting	O
with	O
a	O
line	O
segment	O
of	O
arbitrary	O
length	O
,	O
drawing	O
two	O
shorter	O
segments	O
at	O
right	O
angles	O
to	O
the	O
first	O
through	O
its	O
endpoints	O
,	O
and	O
continuing	O
in	O
the	O
same	O
vein	O
,	O
reducing	O
(	O
dividing	O
)	O
the	O
length	O
of	O
the	O
line	O
segments	O
drawn	O
at	O
each	O
stage	O
by	O
.	O
</s>
<s>
A	O
variant	O
of	O
this	O
construction	O
could	O
also	O
be	O
defined	O
in	O
which	O
the	O
length	O
at	O
each	O
iteration	O
is	O
multiplied	O
by	O
a	O
ratio	O
less	O
than	O
,	O
but	O
for	O
this	O
variant	O
the	O
resulting	O
shape	O
covers	O
only	O
part	O
of	O
its	O
bounding	O
rectangle	O
,	O
with	O
a	O
fractal	B-Application
boundary	O
.	O
</s>
<s>
An	O
alternative	O
process	O
that	O
generates	O
the	O
same	O
fractal	B-Application
set	O
is	O
to	O
begin	O
with	O
a	O
rectangle	O
with	O
sides	O
in	O
the	O
ratio	O
,	O
and	O
repeatedly	O
bisect	O
it	O
into	O
two	O
smaller	O
silver	O
rectangles	O
,	O
at	O
each	O
stage	O
connecting	O
the	O
two	O
centroids	O
of	O
the	O
two	O
smaller	O
rectangles	O
by	O
a	O
line	O
segment	O
.	O
</s>
<s>
The	O
H	B-Algorithm
tree	I-Algorithm
is	O
a	O
self-similar	O
fractal	B-Application
;	O
its	O
Hausdorff	O
dimension	O
is	O
equal	O
to	O
2	O
.	O
</s>
<s>
The	O
points	O
of	O
the	O
H	B-Algorithm
tree	I-Algorithm
come	O
arbitrarily	O
close	O
to	O
every	O
point	O
in	O
a	O
rectangle	O
(	O
the	O
same	O
as	O
the	O
starting	O
rectangle	O
in	O
the	O
constructing	O
by	O
centroids	O
of	O
subdivided	O
rectangles	O
)	O
.	O
</s>
<s>
In	O
VLSI	O
design	O
,	O
the	O
H	B-Algorithm
tree	I-Algorithm
may	O
be	O
used	O
as	O
the	O
layout	O
for	O
a	O
complete	O
binary	O
tree	O
using	O
a	O
total	O
area	O
that	O
is	O
proportional	O
to	O
the	O
number	O
of	O
nodes	O
of	O
the	O
tree	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
H	B-Algorithm
tree	I-Algorithm
forms	O
a	O
space	O
efficient	O
layout	O
for	O
trees	O
in	O
graph	O
drawing	O
,	O
and	O
as	O
part	O
of	O
a	O
construction	O
of	O
a	O
point	O
set	O
for	O
which	O
the	O
sum	O
of	O
squared	O
edge	O
lengths	O
of	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
tour	I-Algorithm
is	O
large	O
.	O
</s>
<s>
The	O
planar	O
H	B-Algorithm
tree	I-Algorithm
can	O
be	O
generalized	O
to	O
the	O
three-dimensional	O
structure	O
via	O
adding	O
line	O
segments	O
on	O
the	O
direction	O
perpendicular	O
to	O
the	O
H	B-Algorithm
tree	I-Algorithm
plane	O
.	O
</s>
<s>
The	O
resultant	O
three-dimensional	O
H	B-Algorithm
tree	I-Algorithm
has	O
Hausdorff	O
dimension	O
equal	O
to	O
3	O
.	O
</s>
<s>
The	O
planar	O
H	B-Algorithm
tree	I-Algorithm
and	O
its	O
three-dimensional	O
version	O
have	O
been	O
found	O
to	O
constitute	O
artificial	O
electromagnetic	O
atoms	O
in	O
photonic	O
crystals	O
and	O
metamaterials	O
and	O
might	O
have	O
potential	O
applications	O
in	O
microwave	O
engineering	O
.	O
</s>
<s>
The	O
H	B-Algorithm
tree	I-Algorithm
is	O
an	O
example	O
of	O
a	O
fractal	B-Algorithm
canopy	I-Algorithm
,	O
in	O
which	O
the	O
angle	O
between	O
neighboring	O
line	O
segments	O
is	O
always	O
180	O
degrees	O
.	O
</s>
<s>
In	O
its	O
property	O
of	O
coming	O
arbitrarily	O
close	O
to	O
every	O
point	O
of	O
its	O
bounding	O
rectangle	O
,	O
it	O
also	O
resembles	O
a	O
space-filling	B-Algorithm
curve	I-Algorithm
,	O
although	O
it	O
is	O
not	O
itself	O
a	O
curve	O
.	O
</s>
<s>
Topologically	B-Architecture
,	O
an	O
H	B-Algorithm
tree	I-Algorithm
has	O
properties	O
similar	O
to	O
those	O
of	O
a	O
dendroid	O
.	O
</s>
<s>
However	O
,	O
they	O
are	O
not	O
dendroids	O
:	O
dendroids	O
must	O
be	O
closed	O
sets	O
,	O
and	O
H	B-Algorithm
trees	I-Algorithm
are	O
not	O
closed	O
(	O
their	O
closure	O
is	O
the	O
whole	O
rectangle	O
)	O
.	O
</s>
<s>
Variations	O
of	O
the	O
same	O
tree	O
structure	O
with	O
thickened	O
polygonal	O
branches	O
in	O
place	O
of	O
the	O
line	O
segments	O
of	O
the	O
H	B-Algorithm
tree	I-Algorithm
have	O
been	O
defined	O
by	O
Benoit	O
Mandelbrot	O
,	O
and	O
are	O
sometimes	O
called	O
the	O
Mandelbrot	B-Algorithm
tree	I-Algorithm
.	O
</s>
