1. B-Tree of degree 3 t=3 data internal nodes 40 80 10 27 31 52 65 88 9628 30 45 50 70 74 77 82 84 97 99 15 25 33 35 53 55 58 60 62 90 92 94
data

2. For simplicity, assume each key links to some associated data40 80 10 27 31 52 65 88 96 3 7 45 50 15 25 82 84 97 99 53 55 58 60 62 28 30 33 35 70 74 77 90 92 94
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3. Bt is a B-tree with minimum degree t (or order m , where m = 2t):
The root has n keys with 1  n  2t-1
Each non-root node has n keys with t-1  n  2t-1
Each non-leaf node with n keys has n+1 children
All leaves are of the same height
For each node as shown in the following k1 k2 ki
ki+1 kn
where
k1 < k2 < ... < ki < ki+1 < ... < kn
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4. Bt is a B-tree with minimum degree t:
and if the node is not a leaf, then k1 k2 ki
ki+1 kn C0 C1 C2 Ci
Ci+1
Cn-1 Cn
ki  min(Ci)  max(Ci)  ki+1, for 1  i  n-1
max(C0)  k1
kn  min(Cn)
maximum and minimum keys in Ci
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5. Bt is a B-tree with minimum degree t:
Some Terminologies ki
ki+1
Ci-1 Ci
the left-child of key
the right-child of key ki ki
We say that a node is full if it has 2t-1 keys
If t=2, B2 is known as tree
(named after the number of children)
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6. Algorithm for inserting a new key (data) into Bt
The new arrived key will be inserted into some leaf
Search the leaf with keys that can contain the new key (according to the key values).
During the search, if we reach a full node, i.e., it already has 2t-1 keys, then we split the node and repeat 1 and 2 again until the new key is successfully inserted.
Note: after splitting a leaf (if necessary), we should have a leaf with room for the new key.
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7. Algorithm for splitting a full rootk1
kt-1 kt
kt+1
k2t-1 C0
Ct-2
Ct-1 Ct
Ct+1
C2t-1
Split 
new root kt k1
kt-1
kt+1
k2t-1 C0
Ct-2
Ct-1 Ct
Ct+1
C2t-1
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8. Algorithm for splitting a full non-root node I. Splitting node
parent node k’ ka
ka+1
k’’
link before splitting kt k1
kt-1
kt+1
k2t-1 C0
Ct-2
Ct-1 Ct
Ct+1
C2t-1
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9. Algorithm for splitting a full non-root node II. Modifying the parent
parent node k’ ka
ka+1
k’’ kt k1
kt-1
kt+1
k2t-1 C0
Ct-2
Ct-1 Ct
Ct+1
C2t-1
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10. Algorithm for splitting a full non-root node II. Modifying the parent
parent node k’ ka kt
ka+1
k’’ k1
kt-1
kt+1
k2t-1 C0
Ct-2
Ct-1 Ct
Ct+1
C2t-1
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11. t=2, B2 : 2-3-4 tree. The root has n keys with 1  n  2t-1
Each non-root node has n keys with t-1  n  2t-1
t=2, B2 : tree. 25 10 20 30 40 25 15 16 20 10 30 40 40 40 20 10 20 30 15 16 10 30 20 10 25 30 40
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12. t=2 B2 : 2-3-4 tree. The root has n keys with 1  n  2t-1
Each non-root node has n keys with t-1  n  2t-1 35
t=2 B2 : tree. 20 10 15 16 25 30 40 35 20 30 20 30 10 15 16 25 35 40 10 15 16 25 40
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13. t=2, B2 : 2-3-4 tree. The root has n keys with 1  n  2t-1
Each non-root node has n keys with t-1  n  2t-1
t=2, B2 : tree. 5 20 30 10 15 16 25 35 40 5
need to split again 5 20 15 20 30 30 15 10 16 25 35 40 10 16 25 35 40
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