Math 55: Discrete Mathematics
UC Berkeley, Spring 2009
Solutions to Homework # 12 (due May 4)
§
10.1, # 3: a) The root is
a
.
b) The internal vertices are
a, b, c, d, f, h, j, q, t
c) The leaves are
e, g, i, k, l, m, n, o, p, r, s, u
.
d) The children of
j
are
q
and
r
.
e) The parent of
h
is
c
.
f) The only sibling of
o
is
f
.
g) The ancestors of
m
are
a, b, f
.
h) The descendents of
b
are
e, f, l, m, n
.
§
10.1, # 13*
a) There are 3 non-isomorphic unrooted trees with Fve vertices: the
star tree, the chain, and one other tree.
b) The three unrooted trees above can be rooted in 2 + 3 + 4 = 9
non-isomorphic ways.
§
10.1, # 18: By Theorem 4(ii), the answer is
mi
+ 1 = 5
·
100 + 1 = 501.
§
10.1, # 22: The model is a full 5-ary tree. There are 10
,
000 internal vertices,
corresponding to the people who send out the letter. By Theorem
4(ii), the total number of vertices is
mi
+ 1 = 50
,
001. Everyone expect
the root receives the letter, so 50
,
000 people receive the letter. There
are 40