B-Trees

CS321 Spring 2019

Steve Cutchin

2

B-Tree Motivation

• When data is too large to fit in main memory, then

the number of disk accesses becomes important.

• A disk access is unbelievably expensive compared

to a typical computer instruction (mechanical

limitations).

• One disk access is worth about 200,000

instructions.

• The number of disk accesses will dominate the

running time.

3

Motivation Cont..

• Secondary memory (disk) is divided into equal-

sized blocks (typical sizes are 512, 2048, 4096 or

8192 bytes)

• The basic I/O operation transfers the contents of

one disk block to/from main memory.

• Our goal is to devise a multiway search tree that

will minimize file accesses (by exploiting disk

block read).

4

m-ary Trees

• A node contains multiple keys.

• Order of subtrees is based on parent node’s keys

• If each node has m children & there are n keys

then the average time taken to search the tree is

logmn.

Etc.

K1 K2 K3 K4

T1 T2 T3

K < K1 K1 < K < K2

B Tree Definition

• A B-Tree is a search tree with a root node.

• Each node in a B-Tree can have multiple keys.

• Each node in a B-Tree can have multiple children.

• The number of children is dependent on the

number of keys.

• A node in a B-Tree has at most 1 more child than

it has keys.

5

Layout of a B-Tree

6

Each node has at most 3 keys and 4 children.

Each node has a minimum of 2 children.

This is a 2-3-4 B-Tree

Important Metrics

• The minimal degree of a B-Tree is defined as:

– Degree = t, t >= 2.

– Every node except root has at least t children.

– Every node except root has at least t-1 keys.

– Every node except root has at most 2*t – 1 keys.

• The order of a B-Tree is defined as:

– Order = m

– No node may have more than m children.

• Therefore: Order = 2*degree; 7

Layout of a B-Tree

8

What is the degree of this B-Tree?

What is the order of this B-Tree?

Size of B Trees

• All leaves in a tree have the same depth.

• The depth of a B-Tree is uniform and equal to its

height.

• By definition all B-Trees are balanced.

9

Size of B Trees

• For a given B-Tree with n keys and degree t

• Height h <= logt((n+1)/2);

• For a given B-tree with height of h and degree t

• n >= 2 * th - 1

10

B-Tree and Block Size

• A B-Tree Node is usually the size of a Disk Page.

• So if a Disk Page = 4096 bytes we want our Node

to be that size:

• Say, 84 bytes overhead for the Node.

• 4 Bytes for each key. 4 Bytes for each child

pointer. 4 bytes for num keys, 4 bytes num

children.

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B-Tree and Block Size

• 4096 = 4K + 4C + 4 + 4 + 84.

• C = K+1.

• 4096 = 4K + 4K+4 + 4 + 4 + 84.

• 4096 = 8K + 12 + 84

• 4096 -12 -84 = 8K

• K = 500 Keys per Node for one block.

• C = 501 Children per Node for each block.

• A tree of height 2 has 125,751,500 Keys

• A tree of height 2 has 251,503 Disk Blocks.

12

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Definition of a B-Tree

• Def: B-tree of degree t is a tree with the following

properties.:

– The root has at least 2 children, unless it is a leaf.

– Every non-root node must have t-1 keys.

– Every non-root internal node has t children.

– If the tree is non-empty the root has at least one key.

– Every node may have at most 2t-1 keys.

– An internal node may have at most 2t children.

– A full tree occurs when every node has 2t-1 keys.

Components of B-Tree Nodes

• Every node x has the following attributes:

– X.n = the number of keys in X

– X.keys[n] = the actual keys.

– X.leaf = is this a leaf? Can the root be a leaf?

– X.child[n+1] = array of pointers to the children.

• Rule: key[1] <= key[2] <= … key[n].

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15

Definition of a B-Tree

• Def: B-tree of order m is a tree with the following

properties:

– The root has at least 2 children, unless it is a leaf.

– No node in the tree has more then m children.

– Every node except for the root and the leaves have at

least m/2 children.

– All leaves appear at the same level.

– An internal node with k children contains exactly k-1

keys.

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B-Trees & Efficiency

• Used in Mac, NTFS, OS2 for file structure.

• Allow insertion and deletion into a tree structure,

based on logmn property, where m is the order of

the tree.

• The idea is that you leave some key spaces open.

So an insert of a new key is done using available

space (most cases).

– Less dynamic then our typical Binary Tree

– Efficient for disk based operations.

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2-3 Trees

G

I | M

J | K

C

D | E A H N | O

B Tree Operations (adt)

• Search(key)

• Insert(key)

• Delete(key)

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Searching m-ary Trees

A generalized SOT will visit all keys in ascending

order.

for (i==1;i<=m-1;i++) {

visit subtree to left of ki

visit ki

}

visit subtree to right of km-1

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Basic Recursive Search

Ordered Recursive Search. Array indexed by 1.

Search(T,k)

for (i==1;i<=m-1;i++) {

if (k < ki) return Search(T.child[i],k);

}

Return Search(T.child[m],k);

Notice the for loop! O(?)

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Insertion

• Insert ki into B-tree of order m.

- We find the insertion point (in a leaf) by doing a search.

- If there is room then enter ki.

- Else, promote the middle key to the parent & split the

node into nodes around the middle key.

• If the splitting backs up to the root, then

– Make a new root containing the middle key.

• Note: the tree grows from the leaves, balance is

always maintained.

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I | K | M

J H N | O L

G | K

M

L

C

D | E A H N | O

I

J

G

I | M

J | K

C

D | E A H N | O

K is promoted again, this

gives the new tree:

L is inserted into

the above tree.

Insertion Example

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Splitting Nodes

T3 T2

A | B | C

T1 T4

T4

C A

B

T1 T2 T3

• Middle key is promoted

• Creating a new root

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Deletion

• If the entry to be deleted is not in a leaf, swap it

with its successor (or predecessor) under the

natural order of the keys. Then delete the entry

from the leaf.

• If leaf contains more than the minimum number of

entries, then one can be deleted with no further

action.

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Deletion Example 1

C

D | E A

C

E A

Delete D

C

D | E A

Delete C D

E A

Successor is promoted, Element D

C is Deleted.

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Deletion Cont...

• If the node contains the minimum number of

entries, consider the two immediate siblings of the

parent node:

• If one of these siblings has more than the

minimum number of entries, then redistribute one

entry from this sibling to the parent node, and

one entry from the parent to the deficient node.

– This is a rotation which balances the nodes

– Note: all nodes must comply with minimum entry

restriction.

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Deletion Example 2

C

D |

E

A

C

D |

E

C

Delete A

C |

D

E

D

E

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Deletion Cont...

• If both immediate siblings have exactly the

minimum number of entries, then merge the

deficient node with one of the immediate sibling

node and one entry from the parent node.

• If this leaves the parent node with too few entries,

then the process is propagated upward.

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Deletion Example 3 G | K

M

L

C

D | E A H N | O

I

J

G | K

M

L

C

D | E A N | O

I

J

Delete H

G | K

M

L

C

D | E A N | O I | J

Combine with parent and 1 sibling of

parent

Node is

deficient

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Deletion Example 3 Cont.. G | K

M

L

C

D | E A N | O I | J

Node is now

deficient

G

L

C

D | E A N | O I | J

K | M

Deficient node is combined with

1 key from parent and sibling of

parent

Node G is legal so propagation

up the tree stops.

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Review of Deletions

• All Deletions take place in leaf nodes

– To delete a internal key swap it with its successor or

predecessor which is a leaf.

– Then Delete

• Deficient Nodes are legalized by:

– Rotation with a sibling and parent.

OR

– Combining with key from parent and sibling

– Propagating up the tree until a legal node is

encountered.

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End Notes

Studies have shown that on average there is about

1/((m/2) -1) splits per insertion.

– E.g.

• For a 2/3 tree there is 1

• For a 10-ary tree there is 1/4

Acknowledgement Dave Bockus

© Dave Bockus

Acknowledgements to:

Dr Frederic Maire Brisbane, Queensland, AUSTRALIA

for some of the material found in this presentation