6.042/18.062J
Mathematics
for
Computer
Science
October
5,
2010
Tom
Leighton
and
Marten
van
Dijk
Problem
Set
5
Readings:
Section
5.4
to
5.7
and
6.16.2.
Problem
1.
[20
points]
Recall
that
a
tree
is
a
connected
acyclic
graph.
In
particular,
a
single
vertex
is
a
tree.
We
define
a
Splitting
Binary
Tree
,
or
SBTree
for
short,
as
either
the
lone
vertex,
or
a
tree
with
the
following
properties:
1.
exactly
one
node
of
degree
2
(called
the
root).
2.
every
other
node
is
of
degree
3
or
1
(called
internal
nodes
and
leaves,
respectively).
For
the
case
of
one
single
vertex
(see
above),
that
vertex
is
considered
to
be
a
leaf.
It
is
easier
to
understand
the
definition
visually,
so
an
example
is
shown
in
Figure
1.
An
example
of
a
tree
which
is
not
an
SBTree
is
shown
in
Figure
2.
(a)
[10
pts]
Show
if
an
SBTree
has
more
than
one
vertex,
then
the
induced
subgraph
ob
tained
by
removing
the
unique
root
consists
of
two
disconnected
SBTrees.
You
may
assume
that
by
removing
the
root
you
obtain
two
separate
connected
componenents,
so
all
you
need
to
prove
is
that
those
two
components
are
SBTrees.
(b)
[10
pts]
Prove
that
two
SBTrees
with
the
same
number
of
leaves
must
also
have
the
same
total
number
of
nodes.
Hint:
As
a
conjecture,
guess
an
expression
for
the
total
number
of
nodes
in
terms
of
the
number
of
leaves
N
(
l
)
.
Then
use
induction
to
prove
that
it
holds
for
all
trees
with
the
same
l
Problem
2.
[20
points]
In
”Die
Hard:
The
Afterlife”,
the
ghosts
of
Bruce
and
Sam
have
been
sent
by
the
evil
Simon
on
another
mission
to
save
midtown
Manhattan.
They
have
been
told
that
there
is
a
bomb
on
a
street
corner
that
lies
in
Midtown
Manhattan,
which
Simon
defines
as
extending
from
41st
Street
to
59th
Street
and
from
3rd
Avenue
to
9th
Avenue.
Additionally,
the
code
that
they
need
to
defuse
the
bomb
is
on
another
street
corner.
Simon,
in
a
good
mood,
also
tosses
them
two
carrots:
He
will
have
a
helicopter
initially
lower
them
to
the
street
corner
where
the
bomb
is.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
Problem
Set
5
A
E
F
D
B
C
G
Figure
1:
Splitting
Binary
Tree:
Node
A
is
the
root,
B
and
E
are
internal
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Graph Theory, Glossary of graph theory, Hamiltonian path, Midtown Manhattan

Click to edit the document details