Lec 28 B Tree Operations

8/3/2019 Lec 28 B Tree Operations

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B-Tree Operations


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Multi way search trees(of order

m)A generalization of Binary Search Trees.

Each node has at mostm

children.

Ifk m is the number of children, then the node

has exactly k-1 keys.

The tree is ordered.


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B-tree of order 5 This means that (other then the root node)

all internal nodes have at least ceil(5 / 2) =

ceil(2.5) = 3 children and hence at least 2keys.

Of course, the maximum number ofchildren that a node can have is 5 (so that4 is the maximum number of keys).

Each leaf node must contain at least 2keys.


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Insertion

In B-Tree (Order 5)

We want to insert C N G A H E K Q M F W L T Z D P R X Y S

Order 5 means that a node can have a

maximum of 5 children and 4 keys.

All nodes other than the root must have a

minimum of 2 keys.

Height of B-Tree grows with splitting only.


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Contd Insert C

Insert N

Insert G

Insert A


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Inserting E, K, and Q We will proceeds without requiring any

splits


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Inserting

Mreq

uires a sp

lit

Note that M happens to be the median key

and so is moved up into the parent node.


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Insert F, W, L, and T Need a split?

These are added without needing any

split.


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Insert Z Rightmost leaf must be split. The median

item T is moved up into the parent node.


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Insert D Causes the leftmost leaf to be split.


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Finally

Insert S

4th node splits.

The parent node is full, so it splits


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Delete Operation


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Delete H

H is in a leaf and the leaf has more than

the minimum number of keys, this is easy.


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Delete T


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Delete R

Although R is in a leaf, this leaf does nothave an extra key.

The deletion results in a node with onlyone key, which is not acceptable for a B-tree of order 5.

If the sib

ling node to the immediate

left orright has an extra key, we can then borrow

a key from the parent and move a key upfrom this sibling.


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Contd In our specific case, the sibling to the right

has an extra key.

So, the successor W of S (the last key in

the node where the deletion occurred), is

moved down from the parent, and the X is

moved up. Of course, the S is moved over so that the

W can be inserted in its proper place.


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Contd


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Finally De

lete E

This one causes lots of problems

How?

Although E is in a leaf, the leaf has no extrakeys, nor do the siblings to the immediate rightorleft.

In such a case the leaf has to be combined withone of these two siblings.

This includes moving down the parent's key thatwas between those of these two leaves. In ourexample, let's combine the leaf containing F withthe leaf containing A C. We also move down theD.


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Contd


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Contd Parent node now contains only one key, G. This

is not acceptable.

Immediate left or right that had a spare key, thenwe would again "borrow" a key.

Since in our example we have no way to borrow

a key from a sibling, we must again combine

with the sibling, and move down the M from the

parent.

In this case, the tree shrinks in height by one.


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Final

B-Tree

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Lec 28 B Tree Operations