The document discusses B-trees, which are a type of search tree used to organize data files. B-trees allow each node to have many children, with the number limited only by block size. This improves efficiency over 2-3 trees by keeping the search tree short. The key operations of B-trees are retrieval, insertion, and deletion of records. Variations like B*-trees and B+-trees aim to reduce the number of nodes in the tree and improve traversal efficiency.

1. Chapter 15 B
External Methods –
B-Trees

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B-Trees
• To organize the index file as an external search tree
– Use block numbers for child pointers
• A child pointer value of –1 is used as the null pointer
Figure 15.10a –Figure 15.10a – Blocks organized into a 2-3 tree

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B-Trees
• If the index file is organized into a 2-3 tree
– Each node would contain
• Either one or two index records, each of the form
<key, pointer>
• Three child pointers
Figure 15.10b –Figure 15.10b – A single node of the 2-3 tree

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B-Trees
• An external 2-3 tree is adequate, but an
improvement is possible
• To improve efficiency
– Allow each node to have as many children as possible
• In an external environment, the advantage of keeping a search
tree short far outweighs the disadvantage of performing extra
work at each node
• Block size should be the only limiting factor for the number of
children

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B-Trees
• Binary search tree
– If a node N has two children, it must contain one key
value
• 2-3 tree
– If a node N has three children, it must contain two key
values
• General search tree
– If a node N has m children, it must contain m – 1 key
values

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B-Trees
Figure 15.11Figure 15.11
a) A node with two children; b) a node with three children; c) a node with m children

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B-Trees
• B-tree of degree m
– All leaves are at the same level
– Nodes
• Each node contains between m – 1 and m/2 records
• Each internal node has one more child than it has records
• Exception: The root can contain as few as one record and can
have as few as two children
– Example
• A 2-3 tree is a B-tree of degree 3
– Each node contains between (3-1) = 2 and 3/2 = 1 records.

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B-Trees
Figure 15.13Figure 15.13
A B-tree of degree 5

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B-Trees
• Retrieval
– Generalized search tree retrieval
• Insertion into a B-tree
– Step 1: Insert the data record into the data file
– Step 2: Insert a corresponding index record into
the index file

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B-Trees
Figure 15.14a and bFigure 15.14a and b
The steps for inserting 55

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B-Trees
Figure 15.14c-eFigure 15.14c-e
The steps for inserting 55

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B-Trees
• Deletion from a B-tree
– Step 1: Locate the index record in the index file
and delete it from the index file
– Step 2: Delete the data record from the data file

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B-Trees
Figure 15.15a and bFigure 15.15a and b
The steps for deleting 73

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B-Trees
Figure 15.15cFigure 15.15c
The steps for deleting 73

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B-Trees
Figure 15.15dFigure 15.15d
The steps for deleting 73

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B-Trees
Figure 14.15e and fFigure 14.15e and f
The steps for deleting 73

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Traversals
• Accessing only the search key of each record, not
the data file
– Not efficiently supported by the hashing
implementation
– Efficiently supported by the B-tree implementation
• The search keys can be visited in sorted order by using an
inorder traversal of the B-tree
• Accessing the entire data record
– Not efficiently supported by the B-tree implementation

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Multiple Indexing
• Advantage
– Allows multiple data organizations
• Disadvantage
– More storage space
– Additional overhead for updating each index
whenever the data file is modified

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Multiple Indexing
Figure 15.16Figure 15.16
Multiple index files

20. Chi-Cheng Lin, Winona State University 20
Variations of B-Trees
• The fewer nodes are in a B-tree, the better.
(Why?)
• Problems of B-tree:
– It could be only half full
• More nodes required
• Space wasted
– Inorder traversal “jumps” around nodes
• B*
-Trees: introduced by Donald Knuth
• B+
-Trees: introduced by H. Wedekind

21. Chi-Cheng Lin, Winona State University 21
B*
-Trees
• B*
-tree of degree m
– All leaves are at the same level
– Nodes
• Each node contains between m – 1 and (2m – 1)/3 records
• Each internal node has one more child than it has records
• Exception: The root can contain as few as one record and as
many as (2m-2) children
– Example
• B*
-tree of degree 9
– Each node contains between (9 – 1) = 8 and (2×9 – 1)/3 = 5
records.

22. Chi-Cheng Lin, Winona State University 22
B*
-Trees
• Splitting nodes
– When a node overflows, it is not split right away
– A split is delayed by redistributing the keys between a
node and its sibling
– When both a node and its sibling are full, an insertion
to the node will split the two nodes into three nodes
• A B*-Tree is always two-third full instead of half
full  number of nodes in the tree is reduced

23. Chi-Cheng Lin, Winona State University 23
B*
-Trees

24. Chi-Cheng Lin, Winona State University 24
B+
-Trees
• References to data are only made from the
leaves
– All index records can be found from the leaves
• Two sets of nodes
– Index set
• Internal nodes
• Provides fast access of data
– Sequence set
• Leaves, linked sequentially
• Provides efficient inorder traversal

25. Chi-Cheng Lin, Winona State University 25
B+
-Trees