AMS 301.3 (Fall, 2009)
Estie Arkin
Exam 2 – Solution sketch
Mean 81.2, median 84.5, high 100 (4 of them!), low 25.
1. (5 points) Let
T
be a 6ary tree with 100 internal nodes. How many leaves does the tree have?
(A correct guess with no work shown will recieve very partial credit.)
n
=
mi
+ 1,
m
= 6,
i
= 100,
so
n
= 601, and
l
=
n

i
= 501.
2. (15 points) Consider the following graph.
A
B
C
D
E
F
8
1
2
3
4
5
6
7
9
11
12
7
(a).
Highlight the edges of a minimum spanning tree of the graph.
Using Kruskal’s algorithm,
egdes would be put into the MST in the following order: (B,E) (C,D) (D,E) (B,A) (E,F). The cost
of this tree is 20.
A
B
C
D
E
F
1
2
3
7
7
(b). The cost of every edge in the graph is multiplied by 10. Circle the correct answers:
Does this change the edges in the minimum spanning tree? NO, Kruskal’s algorithm would put the
edges in the tree in the same order.
Does this change the cost of the minimum spanning tree? YES, the new minimum cost is 10 times
the old min cost.
(c). Ignoring the costs on the edges, show the BFS tree of the graph starting at node A.
A
B
C
D
E
F
Common mistake: drawing the MST again, or doing DFS.
3.
(8 points) We wish to model the word problem as a graph problem: The Happy Gardner, a
lawnmowing service, has been contracted to mow the lawns at 15 houses today.
The gardener
would like to plan his schedule for the day, namely, in what order the will the different lawns be
1
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mowed. We may assume the gardener starts his day at his house, and would like to return home
as soon as possible. We also assume travel times between houses are known.
(a). What do the nodes of the graph to be constructed represent? The houses (15 to be mowed
and gardnere’s house).
(b).
What do the edges of the graph to be constructed represent?
Travel between 2 houses.
Common mistake: saying edges are times or distances to travl.
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 Spring '08
 ARKIN
 Graph Theory, Shortest path problem, Planar graph, shortest path tree

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