csi35chapter 11 review - natna.info · 11 csi35chapter 11 review 5. a) how many edges does a tree...

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CSI35 Chapter 11 Review

1. Which of the graphs are trees?

a b

c d

e f

a b

c

d

e f

a bc

d

e f

x y

q r

z p

g

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1. Which of the graphs are trees?

a b

c d

e f

a b

c

d

e f

a bc

d

e f

x y

q r

z p

g

3


2. Answer the questions about the following treea

b c d e

f

1) Which vertex is the root of the tree?2) what is the height of the tree?3) Which vertices are children of h?4) Which vertices are siblings of n?

5) What vertices are internal?6) Which vertices are leaves?7) Which vertex is parent of u ?8) Which vertices are ancestors of t ?9) Which vertices are descendants of f ?10) what vertices are at level 3?11) Is the tree balanced?

g h i j k m

n o p q r st u v

w x yz

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2. Answer the questions about the following treea

b c d e

f

1) Which vertex is the root of the tree? a2) what is the height of the tree? 43) Which vertices are children of h? none4) Which vertices are siblings of n? 0 and p

5) What vertices are internal? a, b, c, d, e, f, I, j, m, n, q.6) Which vertices are leaves? g, h, k, o, p, r, s, t, u, v, w, x, y, z, , 7) Which vertex is parent of u ? m8) Which vertices are ancestors of t ? m, e, a9) Which vertices are descendants of f ? n, o, p, w, x10) what vertices are at level 3? n, o, p, q, r, s, t, u, v11) Is the tree balanced? No, because we have leaves at levels 2,3,and 4

g h i j k m

n o p q r st u v

w x yz

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3. Determine which trees are full, complete, or neither.

a

b c

d e f

h

g

T1 T

2

a

b c

d e

f g

T3

T4

T5

6


3. Determine which trees are full, complete, or neither.

a

b c

d e f

h

g

T1 T

2

a

b c

d e

f g

T3

T4

T5

neitherneither

full

complete complete

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4. a) How many nonisomorphic rooted trees are there with five vertices?

b) How many nonisomorphic unrooted trees are there with five vertices?

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4. a) How many nonisomorphic unrooted trees are there with five vertices?

2 vertices of degree 1, and 3 vertices of degree 2

3 vertices of degree 1, 1 vertex of degree 2, and 1 vertex of degree 3

4 vertices of degree 1, and 1 vertex of degree 4

Answer: 3 trees

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4. a) How many nonisomorphic unrooted trees are there with five vertices?

2 vertices of degree 1, and 3 vertices of degree 2

3 vertices of degree 1, 1 vertex of degree 2, and 1 vertex of degree 3

4 vertices of degree 1, and 1 vertex of degree 4

b) How many nonisomorphic rooted trees are there with five vertices?

Answer: 9 trees

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5. a) How many edges does a tree with 80 vertices have?

b) How many internal vertices does a full binary 3-ary tree with 113 leaves have?

c) How many vertices does a complete 4-ary tree of height 6 have?

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5. a) How many edges does a tree with 80 vertices have?

A tree with n vertices has n-1 edges, hence the given tree has 79 edges. Answer: 79 edges

b) How many internal vertices does a full binary 3-ary tree with 113 leaves have?

m = 3, l = 113, hence

Answer: 56 internal verticesc) How many vertices does a complete 4-ary tree of height

6 have?

i=l−1m−1

=113−13−1

=1122

=56

...

level 0: 1 vertex 40

level 1: 4 vertices 1*4 = 41

level 2: 4 vertices 1*41 = 42 … 46

Answer: 1 + 4 + 42 + 43 + 44+ 45 + 46 vertices.

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6. Given a binary tree (see below), insert the following numbers, one by one: 10, 14, 66, 79, and 37.

45

26 76

65 89

53 70

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6. Given a binary tree (see below), insert the following numbers, one by one: 10, 14, 66, 79, and 37.

45

26 76

65 8910

7053

37

14

66

79

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7. How many weighings of a balance scale are needed to find a counterfeit coin among four coins if the counterfeit coin can be either heavier or lighter than the others?

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7. How many weighings of a balance scale are needed to find a counterfeit coin among four coins if the counterfeit coin can be either heavier or lighter than the others? Describe an algorithm.Answer: 3 weighings. The algorithm:

x1,x

2 : x

3,x

4

x1: x

2

x1,x

2 lighter

x3: x

4

x3,x

4 lighter

x1 x

3x

2x

4

x1lighter x

2 lighter x

3 l x

4 lighter

same

x1: x

3

same

x3

x4

x1: x

3

same

same

x1

x2

not balanced

not balanced

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8. Construct the binary tree with prefix codes representing these coding schemes:a: 1 b: 01 c: 001 d: 0001 e: 00001

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8. Construct the binary tree with prefix codes representing these coding schemes:a: 1 b: 01 c: 001 d: 0001 e: 00001

d

a

b

c

0

0

0

1

1

1

1

A variation of a question: determine which of the given codes are prefix codes.

1e

0

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9. Use Huffman coding to encode the symbols with the given frequencies.a: 0.4 b: 0.2 c: 0.1 d: 0.15 e: 0.13 f: 0.02

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0.02

f c

0.1

e

0.13

d

0.15

b

0.2

a

0.4 Initial step

0.12

fc

0 11st iteration

2nd iteration

e

0.13

d

0.15

b

0.2

a

0.4

0.25

fc

e0 1

0 1

d

0.15

b

0.2

a

0.4

20



4th iteration

3rd iteration

a

0.40.25

fc

e0 1

0 10.35

db

0 1

db

0 1

fc

e0 1

0 1

100.6

a

0.4

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5th iteration

101

db

0 1

fc

e0 1

0 1

0 1a

prefix code:a: 1b: 000c: 0110d: 001e: 010f: 0111

The average number of bits required to encode a symbol is:0.4*1 + 0.2*3 + 0.1*4 + 0.15*3 + 0.13*3 + 0.02*4 = 2.32

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10. List the vertices of the tree visited during aa) preorder traversal b) inorder traversalc) postorder traversalof the tree.

a

b c

e g h

mk

d

n

jif

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10. List the vertices of the tree visited during aa) preorder traversal b) inorder traversalc) postorder traversalof the tree.

preorder traversal: a, b, e, k, m, f, c, g, h, d, i, n, jinorder traversal: k, e, m, b, f, a, g, c, h, n, i, d, jpostorder traversal: k, m, e, f, b, g, h, c, n, i, j, d, a

a

b c

e g h

mk

d

n

jif

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10. Represent (A B) – (A (B – A)) using an ordered rooted tree.Then write the expression in prefix, postfix and infix notations.

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10. Represent (A B) – (A (B – A)) using an ordered rooted tree.Then write the expression in prefix, postfix and infix notations.

–

A B A –

B A

prefix : – A B A – B Apostfix: A B A B A – –infix: (A B) – (A (B – A))

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11. Evaluate the prefix expression + - ^ 3 2 ^ 2 3 / 6 – 4 2 .

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11. Evaluate the prefix expression + - ^ 3 2 ^ 2 3 / 6 – 4 2 .

+ - ^ 3 2 ^ 2 3 / 6 – 4 2 4-2= 2+ - ^ 3 2 ^ 2 3 / 6 2 6/2 = 3+ - ^ 3 2 ^ 2 3 3

23 = 8+ - ^ 3 2 8 3 32 = 9+ - 9 8 3 9 – 8 = 1+ 1 31 + 3 = 4

+ - ^ 3 2 ^ 2 3 / 6 – 4 2 = 4

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12. Use depth-first-search (DFS) algorithm to produce a spanning tree for the given simple graph. Use vertex a as the root of the spanning tree.

bc g

a

jd

i

e

h

f

ko

m

n

p

q

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bc g

a

jd

i

e

h

f

ko

m

n

p

q

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bc g

a

jd

i

e

h

f

ko

m

n

p

q

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13. Use breadth-first-search (BFS) algorithm to produce a spanning tree for the given simple graph. Use vertex a as the root of the spanning tree.

bc g

a

jd

i

e

h

f

ko

m

n

p

q

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bc g

a

jd

i

e

h

f

ko

m

n

p

q

L = {a, b, c, d, e, f, i, g, h, j, k, q, m, n, o, p}

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bc g

a

jd

i

e

h

f

ko

m

n

p

q

L = {a, b, c, d, e, f, i, g, h, j, k, q, m, n, o, p}

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14. Use Prim's or Kruskal's algorithms to find a minimum spanning tree for the given weighted graph.

b c

g

a

j

d

i

e hf

k

o

m

n p q3

3

3 3

2 2

2

2

2

2

2 2

4

1 2 1

2

2

1

32

1

3

3 3

3 4

4 3

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b c

g

a

j

d

i

e hf

k

o

m

n p q3

3

3 3

2 2

2

2

2

2

2 2

4

1 2 1

2

2

1

32

1

3

3 3

3 4

4 3

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b c

g

a

j

d

i

e hf

k

o

m

n p q

3 3

2 2

2

2 2

2 2

1 1

2

2

1

2

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csi35chapter 11 review - natna.info · 11 csi35chapter 11 review 5. a) how many edges does a tree...

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