1. Unions

2. Which of the following are equivalence relations?
Equality
Equivalent +/- error
“has the same absolute value”
The empty relation (nothing equivalent)

3. Which of the following are equivalence relations?
Equality (yes)
Equivalent +/- error (no)
“has the same absolute value” (yes)
The empty relation (nothing equivalent) (no)

4. Unions by size create on set containing numbers between 1 and 10
What does find 1 return? 1 2 3 4 5 6 7 8 9 10

5. Unions by size How will the tree look after Join(2, 4)? 1 2 3 4 5 6 78 9 10

6. Unions by size How will the tree look after Join(2, 4)?
What about after Join(2, 6)? 2 1 3 4 5 6 7 8 9 10

7. Unions by size How will the tree look after Join(2, 6)?
What about after Join(2, 8)? 2 1 3 4 5 6 7 8 9 10

8. Unions by size
Why aren’t find and union O(1)? 2 1 3 4 5 6 7 8 9 10

9. Unions by size
What happens after Union(3, 2)? 3 2 1 4 5 6 7 8 9 10

10. Unions by size What happens after Union(3, 2)?
If we made all 2’s children point to 3, what would be the cost? 3 2 1 4 5 6 7 8 9 10

11. B-Trees
Is the following a 3-5 B-Tree? If not, what would the correct tree be?
(Assume leaves can have at most 3 keys) 7 8 9 11

12. B-Trees
Not a 3-5 B-Tree
Correction 7 8 9 11 8 9 11

13. B-Trees
Is the following a 3-5 B-Tree? If not, what would be the correct tree?
(Assume leaves can have at most 3 keys) 7 10 4 5 11 12 8 9

14. B-Trees
What happens if we delete 11? 7 10 4 5 11 12 8 9

15. B-Trees
What happens if we delete 11? 7 10 4 5 12 8 9

16. B-Trees
What happens if we delete 11? 7 4 5 8 9 12