Math 184A
Second Exam (50 points)
2 March 2005
•
Please put your name and ID number on your blue book.
•
The exam is CLOSED BOOK except one 2sided page of
notes.
•
Calculators are NOT allowed.
•
You must show your work to receive credit.
1. (10 pts.) Let
G
be the simple graph with vertex set
{
0
,
1
,
2
,
3
,
4
,
5
}
and edges
braceleftbig
{
0
,
1
}
,
{
0
,
2
}
,
{
0
,
3
}
,
{
0
,
4
}
,
{
0
,
5
}
,
{
1
,
2
}
,
{
3
,
4
}
bracerightbig
.
Sketch
G
and compute
P
G
(
x
), the chromatic polynomial of
G
.
2. (12 pts.)
The local description of a decision tree for constructing sequences of A’s
and B’s is given below. The notation BA
S
(
n

2) means place BA in front of each
sequence produced by
S
(
n

2).
S
(1)
A
B
S
(2)
A
S
(1)
BA
S
(
n
) (
n
≥
3)
A
S
(
n

1)
BA
S
(
n

2)
Let
S
*
(
n
) denote the entire decision tree. Thus
S
*
(1) =
S
(1) and
S
*
(2) has the three
leaves AA, AB, and BA.
(a) Prove that each leaf of
S
*
(
n
) is an
n
long sequence of A’s and B’s.
(b) Prove that each nonempty sequence of A’s and B’s will be the leaf of some
S
*
(
n
)
if and only if the sequence does not contain two B’s in a row.
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 Math, Plant morphology, Recurrence relation, Generating function, downward edges

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