1. B+Tree Example n=3
Root
100
120
150
180 30 3 5 11
120
130 30 35
100
101
110
180
200
150
156
179 1

2. B+ tree in textbook’s notation n=3
Leaf:
Non-leaf: 30 35 30 35 30 30 2

3. Size of nodes: n+1 pointers n keys
(fixed) 3

4. Don’t want nodes to be too empty
Use at least
Non-leaf: (n+1)/2 pointers
Leaf: (n+1)/2 pointers to data 4

5. B+tree rules tree of order n
(1) All leaves at same lowest level (balanced tree)
(2) Pointers in leaves point to records except for “sequence pointer”
(to next leaf) 5

6. (3) Number of pointers/keys for B+tree
Max Max Min Min
ptrs keys ptrsdata keys
Non-leaf
(non-root)
n+1 n
(n+1)/2
(n+1)/2- 1
Leaf
(non-root)
n+1 n
(n+1)/2
(n+1)/2
Root
n+1 n 1 1 6

7. Insert into B+tree (a) simple case (insert 32)
space available in leaf
(b) leaf overflow (insert 7)
(c) non-leaf overflow (insert 160)
(d) new root (insert 45) 7

8. Deletion from B+tree (a) Simple case - no example
(b) Coalesce with neighbor (delete 50)
(c) Re-distribute keys (delete 50)
(d) Cases (b) or (c) at non-leaf (delete 37) 8

9. B+tree deletions in practice
Often, coalescing is not implemented
Too hard and not worth it! 9

10. Variation on B+tree: B-tree (no +)
Idea:
Avoid duplicate keys (leaf and non-leaf)
Have record pointers in non-leaf nodes 10

11. K1 P1 K2 P2 K3 P3 to record to record to record
with K1 with K with K3
to keys to keys to keys to keys
< K K1<x<K K2<x<k >k3
K1 P1
K2 P2
K3 P3 11

12. B-tree example n=2 65
125 25 45 85
105
145
165 10 20 30 40 50 60 70 80 90
100
110
120
130
140
150
160
170
180 12

13. B-tree example n=2 sequence pointers not useful now! 65 125 25 45 85
(but keep space for simplicity) 65
125 25 45 85
105
145
165 10 20 30 40 50 60 70 80 90
100
110
120
130
140
150
160
170
180 13

14. Note on inserts Say we insert record with key = 25 10 20 30 n=3 leaf14

15. Note on inserts Say we insert record with key = 25 Afterwards: 10 2030
n=3
leaf 10 20 – 25 30
Afterwards: 15