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Definition assumes external nodes(extended m-way search tree).
B-tree of orderm.
m-way search tree.
Not empty => root has at least 2 children.
Remaining internal nodes (if any) have at
least ceil(m/2) children.External (or failure) nodes on same level.
Definition Of B-Tree
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2-3 And 2-3-4 Trees
B-tree of order m. m-way search tree.
Not empty => root has at least 2 children.
Remaining internal nodes (if any) have atleast ceil(m/2) children.
External (or failure) nodes on same level.
2-3 tree is B-tree of order 3.
2-3-4 tree is B-tree of order 4.
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Insert
15 20
8
4
1 3 5 6 30 409
Insertion into a full leaf triggers bottom-up node
splitting pass.
16 17
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Split An Overfull Node
aiis a pointer to a subtree.
piis a dictionary pair.
m a0 p1 a1 p2 a2pm am
ceil(m/2)-1 a0 p1 a1 p2 a2pceil(m/2)-1 aceil(m/2)-1
m-ceil(m/2) aceil(m/2) pceil(m/2)+1 aceil(m/2)+1pm am
pceil(m/2)plus pointer to new node is inserted in
parent.
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Insert
15 20
8
4
1 3 5 6 30 409
Insert a pair with key = 2.
New pair goes into a 3-node.
16 17
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Insert Into A Leaf 3-node Insert new pair so that the 3keys are in
ascending order.
Split overflowed node around middle key.
1 2 3
Insert middle key and pointer to new node
into parent.
1 3
2
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Insert
15 20
8
4
1 3 5 6 30 409
Insert a pair with key = 2.
16 17
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Insert
15 20
8
4
5 6 30 409 16 17
31
2
Insert a pair with key = 2plus a pointer into parent.
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Insert
Now, insert a pair with key = 18.
15 20
8
1
2 4
5 6 30 409 16 173
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Insert Into A Leaf 3-node Insert new pair so that the 3keys are in
ascending order.
Split the overflowed node.
16 17 18
Insert middle key and pointer to new node
into parent.
1816
17
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Insert
Insert a pair with key = 18.
15 20
8
1
2 4
5 6 30 409 16 173
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Insert
Insert a pair with key = 17plus a pointer into parent.
15 20
8
1
2 4
5 6 30 4093
18
16
17
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Insert
Insert a pair with key = 17plus a pointer into parent.
8
1
2 4
5 6 30 4093 16
17
15
18
20
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Insert
Now, insert a pair with key = 7.
1
2 4
5 6 30 4093 16
15
18
20
8 17
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Insert
Insert a pair with key = 6plus a pointer into parent.
30 401
2 4
93 16
15
18
20
8 17
5
7
6
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Insert
Insert a pair with key = 4plus a pointer into parent.
30 401 93 16
15
18
20
8 17
6
4
2
5 7
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Insert
Insert a pair with key = 8plus a pointer into parent.
There is no parent. So, create a new root.
30 401
93
16
15
18
20
6
8
2
5 7
417
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Insert
Height increases by 1.
30 401 93 16
15
18
2062
5 7
4 17
8