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Root. B+Tree Example n=3. 100. 120 150 180. 30. 3 5 11. 120 130. 180 200. 100 101 110. 150 156 179. 30 35. 1. B+ tree i n textbook’s notation n=3. Leaf: Non-leaf:. 30 35. 30. 35. 30. 30. 2. Size of nodes: n+1 pointers n keys. (fixed). 3.

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  1. Root B+Tree Example n=3 100 120 150 180 30 3 5 11 120 130 180 200 100 101 110 150 156 179 30 35 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. (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) Deletion from B+tree 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. to record to record to record with K1 with K2 with K3 to keys to keys to keys to keys < K1 K1<x<K2 K2<x<k3 >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! • (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 leaf 14

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

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