Trees

Section 6.5

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Trees

Definition

Examples

Two Examples

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3 Topics

1. Minimal Cost Spanning

Trees

2. Counting Trees

3. Searching Trees

Animations

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Minimal Cost Spanning TreeExample: A planned community

Build a road system to begin for a 1000-acre planned community. It will include a convenience store, golf course, tennis courts, and three condo units

The developer has two goals

1.

2.

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Golf

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Condo 1

Condo 3

Tennis

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Best of the 3 animations was $72,000. You can do better. Click when you find one

Got i

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Yes

Yes

Yes

No, loop

Yes

No, loopYes

Stop

$ ,000

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Summarize the Algorithm

1.

2.

3.

Resulting set of edges is called a

minimal cost spanning tree

Joseph Kruskal

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Understanding the terms

“Minimal cost” = Obvious

“Spanning” = No isolated vertices

“Tree” = A graph with no loops

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Continental Cable Vision will have to spend

$405,000 $456,000 $505,000 $528,000 None ofthe ab...

0% 0% 0%0%0%

1. $405,000

2. $456,000

3. $505,000

4. $528,000

5. None of the above

Avon

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Vernon

SouthWindsor

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2. Counting

Two teams, A and B, play a “best-of 5” game series.

For example, here are some ways that A could win:

In how many different ways can A win the series?

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And the answer is…?

6 8 10 12

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1. 6

2. 8

3. 10

4. 12

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A “Tree” Solution

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“Red Sox vs. Yanks”

The Red Sox play the Yanks in an unusual 4-game series. If the visiting team (in this case the beloved Red Sox) wins two games, they win the series. The home team must win three games to win the series

For example:

In how many different ways can the series be played?

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And the answer is…?

1. 4 ways

2. 6 ways

3. 8 ways

4. 10 ways

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A red chip, a white chip, and a blue chip are placed in a container. A person blindly picks one chip, notes its color, and then replaces it in the container. The person then chooses a second chip from the container and notes its color.

Use a tree to determine the number of ways in which exactly one of the two chips drawn is the blue chip1. 2

2. 3

3. 4

4. 5

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3. Searching

You secretly choose an integer from 1-100. I claim that I can find your number after I ask you X “yes/no” questions. What is X?

1. 7-10

2. 11-15

3. 16-20

4. More than 20

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Range # Questions

1 - 1,000,000

1 - 100,000

1 - 10,000

1 - 1,000

1 - 100

End of 6.5

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$0$30,000$57,000$87,000$115,000$131,000

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$1,000’s$0$30,000$53,000$67,000$94,000$122,000

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Units are in

$1,000’s$0$11,000$21,000$35,000$54,000$72,000

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Family Trees

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Zork

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Minimal Cost Spanning Trees

Joseph B. Kruskal 1928 -

Meta - Material

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A tree has ten vertices of degree 2, ten vertices of degree 3, ten vertices of degree 4, one vertex of degree 5, and its remaining vertices have degree 1. How many vertices are in the graph?

1. Matt????

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Cheapest Link

• Like NN start with two closest vertices, but keep on selecting pairs of vertices with the shortest distance between them. Not just the ones adjacent to the already selected pairs.

• Restrictions: don’t complete a circuit until all vertices are included

• A link may not be added if it results in more than two edges coming from the same vertex

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Nearest Insertion

• Start once again with two closest vertices

• Then make a circuit using those two and a third vertex, namely the one that is nearer to either one of the original two vertices

• Now include the unused vertex closest to any of the three.

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Bonus questions

• Jack and Jill were at a party with three other married couples. As people were introduced a good deal of handshaking took place.

• No one shook hands with his or her spouse• No none shook hands with herself or himself• No one shook hands with anyone more than once.

• When leaving the party, Jill asked the other seven people how many hands they had shaken aand received 7 different answers.

• How many hands did Jill shake at the party?• How many handshakes for Jack?

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Ex. 3 Oil Pipeline

A pipeline is to be constructed to take oil across the northern part of Canada

Find the minimum cost to ship the oil between connecting points

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A

B C

G

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6020

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25 3055

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F50

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1 2 3 4 5

And, the answer is …..(Choose the closest answer)

1. $49,000

2. $55,000

3. $59,000

4. $69,000

5. $73,000

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Test question?

If there are N locations (vertices), how many edges are there?

1. N

2. N + 1

3. N – 1

4. 5

6.5 Trees 36

Cable Service

Continental Cable Vision wishes to bring their services to 7 towns in CT. Every pair of towns must be connected by the service. Construct a minimal cost-spanning tree to solve the problem

Avon

Simsbury

Suffield

Bloomfield

Vernon

SouthWindsor

92

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94 80 77

10085

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977

5

100

Granby

Units in $1,000’s

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