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Root

B+Tree Example n=3

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B+ tree in textbook’s notationn=3

Leaf:

Non-leaf:

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Size of nodes: n+1 pointers

n keys (fixed)

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Don’t want nodes to be too empty

• Use at least

Non-leaf: (n+1)/2 pointers

Leaf: (n+1)/2 pointers to data

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B+tree rules tree of order n

(1) All leaves at same lowest level(balanced tree)

(2) Pointers in leaves point to recordsexcept for “sequence pointer”

(to next leaf)

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(3) Number of pointers/keys for B+tree

Non-leaf(non-root) n+1 n (n+1)/2 (n+1)/2- 1

Leaf(non-root) n+1 n

Root n+1 n 1 1

Max Max Min Min ptrs keys ptrsdata keys

(n+1)/2 (n+1)/2

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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)

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(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

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B+tree deletions in practice

– Often, coalescing is not implemented– Too hard and not worth it!

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Variation on B+tree: B-tree (no +)

• Idea:– Avoid duplicate keys (leaf and non-leaf)

– Have record pointers in non-leaf nodes

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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

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B-tree example n=2

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B-tree example n=2

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• sequence pointers not useful now! (but keep space for simplicity)

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Note on inserts

• Say we insert record with key = 25

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leaf

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Note on inserts

• Say we insert record with key = 25

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leaf

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– 20 –

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• Afterwards: