1. Data Structures and Algorithms
B-Trees with Minimum=1
2-3 Trees
Data Structures and Algorithms

2. Data Structures and Algorithms
Delete (T, X, success)
/*Delete from tree T the item with key X. The operation
fails if no such item. The flag success indicates whether the
operation succeeds*/
//Attempt to locate I with search key X
IF I is present THEN
swap item I into leaf L which contains the inorder
successor of I
/* deletion begins from leaf L */
IF L has no items THEN Fix (L)
success := true
ELSE
success := false
Data Structures and Algorithms

3. Fix (N)
/* N is a node with no item. Note, if N is an internal node,
then it has one child. */
Let P be the parent of N. If N is the root, delete it and return.
IF some sibling of N has two items THEN
distribute items among N, the sibling, and P
IF N is internal THEN
move the appropriate child from the sibling to N
ELSE
/* must merge the node */
Choose an adjacent sibling S of N
Bring the appropriate item down from P into S
If N is internal THEN
Move N’s child to S
Delete node N
If P is now without an item THEN Fix (P)

4. Data Structures and Algorithms
Insert (T, newitem)
/* Insert newitem into tree T */
Let X be the search key of new item
Locate the leaf L in which X belongs
Add newitem to L
IF L now has three items THEN
Split (L)
Data Structures and Algorithms

5. Split (N)
/* Split node N which contains 3 items. Note that
if N is internal then it has 4 children */
Let P be the parent of N
/* if N is the root, then create a new node P */
Replace node N by two nodes, N1 and N2
Give N1 the item in N with the smallest search key value
Give N2 the item in N with the largest search key value
If N is an internal node THEN
N1 becomes the parent of N’s two leftmost children
N2 becomes the parent of N’s two rightmost children
Send up to P the item in N with the middle search key value
If P now has 3 items THEN
Split (P)

6. Data Structures and Algorithms
Insertion
Given 50
,90
10,
Insertions
are always
at a leaf
Insert 39 50
,90
10, ,
Data Structures and Algorithms

7. Data Structures and Algorithms
Insert 38 50
,90
10, ,39,
illegal 50
30, ,90
10,
Data Structures and Algorithms

8. Data Structures and Algorithms
Insert 37 50
30, ,90
10, ,
When the height
grows it does so
from the top.
Data Structures and Algorithms

9. Data Structures and Algorithms
Insert 36 50
30, ,90
10, ,37,
illegal
37,50
10, 50
30,37,39
10,
illegal
Data Structures and Algorithms

10. Data Structures and Algorithms
Insert 35, 34, 33
37,50 30
10, ,36
37,50 30
10, ,35,36
37,50
30,35
10,
illegal h
ALL leaves are at the same level
Data Structures and Algorithms

11. Data Structures and Algorithms
37,50
30, ,90
10, , h
Data Structures and Algorithms

12. Data Structures and Algorithms
Deletion
Given 50
,90
10,
Delete 50 60
10, , 60
,90
10,
Data Structures and Algorithms

13. Data Structures and Algorithms
Delete 100 60
10, ,80 60
10,
Data Structures and Algorithms

14. Data Structures and Algorithms
Delete 60 70 30
10, ,90 N 70
10,
10, ,90
30,70
10, ,90
Data Structures and Algorithms

15. Data Structures and Algorithms
Delete 70
30,80
10,
Delete 80
30,90
10, 30
10, ,90
Data Structures and Algorithms

16. Data Structures and Algorithms
Given 50
,90
10,
Delete 70
You always begin deletion from a leaf so swap with inorder successor. 50
,90
10, 50
80,90
illegal
Data Structures and Algorithms

17. Data Structures and Algorithms50 90
60, 50
10, ,
Data Structures and Algorithms

18. Data Structures and Algorithms
Delete 100 50 90
60,80
This leaf can spare a value 50
10, 50 80
Data Structures and Algorithms

19. Data Structures and Algorithms
Delete 80 50
10, 50
10, 50 30
10, ,90
Can‘t spare a value
30,50
10, ,90
Data Structures and Algorithms