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Ceng-112 Data Structures I 2007 1

Chapter 10

MultiwayTrees

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M-Way Trees

M-way tree is a search tree.

Each node of tree can have from zero to m

subtrees,

m is defined as the order of the tree.

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Ceng-112 Data Structures I 2007 3

Figure 10-1

Given the nonempty multiway tree, the following properties are defined:

1. Each node has 0 to m subtrees.

2. Given a node contains k subtree pointers, some of which may be null,and k-1 data entries.

3. The key values in the first subtree are all less than the key in the first entry,

the key values in the other subtrees are all greater than or equal to the key in

their parent entry.

4. The keys of the data entries are ordered.

M-Way Trees

It is not balanced!

m = 4

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M-Way Trees

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Figure 10-4

B-Trees

A B-tree of order; m=5.Min entry : Ceiling|5/2| - 1 = 2 entries.

Max entry: 5 1 = 4 entries.

Min subtrees : Ceiling|5/2|=3

Max subtrees: 5

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Figure 10-5

B-Tree Insertion

B-tree grows from the bottom

up.

When the node is full, this

condition is called asoverflow.

Overflow requires that the

node be split into two nodesand median entry is pushed

up, the tree grows one level.

m=5

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Figure 10-9 (a, b, and c)

B-Tree Insertion

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Figure 10-9 (d, e, and f)

B T B l

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Figure 10-12

B-Tree Balance

Order m=5.

Min entry = 2 entries.

Max entry= 4 entries.

B T C bi

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Figure 10-13

B-Tree Combine

Order m=5

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Figure 10-14 (d and e)

B T T l

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Figure 10-15

B-Tree Traversal

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Simplified B-Trees

There are two types;

1. 2-3 trees; is a B-tree order 3. Each non-root

node has eighter 2 or 3 subtrees. The root

may have zero, two or three subtrees.

2. 2-3-4 trees; is a B-tree order 4. Each node

can have 2, 3 or 4 children.

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Figure 10-16

B-Tree of order 3. (2-3 trees).

Simplified B-Trees

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Figure 10-17

B-Tree of order 4. (2-3-4 trees).It is a B-tree order 4. Each node can have 2, 3 or 4 children.

Simplified B-Trees

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Exercise

Calculate the maximum number of data

entries in a:

1. 3-way tree of heigh 3:

2. 4-way tree of heigh 5:

3. m-way tree of heigh h:

331 = 26

451 = 1023

mh1

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Exercise

Draw the B-tree of order 3 created by inserting the following

data arriving sequence:92 24 6 7 11 8 22 4 5 16 19 20 78

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