1. B-Trees

2. But first, a little note about data structures
Not all data structures work well as file structures
Example: Binary Search Tree
Knight
Gibson
Sanders
Coleman
Hudson
Monroe

3. Motivation for B-Trees
Index too large for memory
search time better than binary search
not just fast search, but also fast delete and fast insert
What's the "B" stand for?
Bayer and McCreight
Boeing
balanced, bushy, broad

4. Indexed Files Searching Deleting Adding Dannelly Duncan Walters
Data File
Indexed Files
Level Two
Key
DRRN
Adams 6 1
Barnes 2
Bell 18 3
Bishop 8 4
Camp 80 5
Carey
Conner 19 7
Critter
Crook 99 9
Dannelly 21 10
Davis 20 11
Dinkins 12
Duncan
. . .
Faulk
Farrow
Foster
Fuller 98
...
West 81
Wilks
Zane
Zinn
Name
Yadda yadda
Carey 1
Foster 2
Barnes 3
Zinn 4
Critter 5
Faulk 6
Adams 7
Wilks 8
Bishop 9
Farrow 10
Duncan 11
Dinkins 12
West
. . . 18
Bell 19
Conner 20
Davis 21
Dannelly
... 80
Camp 81
Zane 98
Fuller 99
Crook
Searching
Dannelly
Deleting
Duncan
Adding
Walters
Aardvark
Level One
Key
IRRN
Adams 1
Davis 10 2
Foster 20 3
Ingram 30 4
Lambert 40 5
Norris 50 6
Randall 60 7
Tyler 70 8
West 80 9
Young 90

5. B-Tree Informal Definition
multi-level indexes
nodes are indexes, indexes are nodes
"Order" - maximum references in a node, minimum references is ½ the order
When node fills, split it and move up largest key
When node is too empty, combine it with parent

6. Example Insertion Insert these letters into an 4-order B-Tree
C S D T A M P I B W N G U R
After C S D and T
After A, node splits and largest keys move up
M and P are added to right node, but so is I C D S T D T A C D S T D P T A C D I M P S T

7. C S D T A M P I B W N G U R B, W, and N are no problem
Insertion of G causes another split, then U is no problem
Inserting R causes right node to split, then root to split D P W A B C D I M N P S T W D M P W A B C D G I M N P S T U W P W D M P T W A B C D N P U W G I M R S T

8. Analysis Order Size? Number of file accesses to Search
match to disk cluster size and memory
Number of file accesses to Search
depth of tree
so bigger the Order the better
Order = 100 and Levels = 4 == 100million records
Number of file accesses to Delete
search downward to the leaf
modify node
if it was largest, adjust parent node

9. Analysis Number of file accesses to Add best case worst case (split)
search downward
adjust the node
worst case (split)
search downward to the leaf
insert, overflow detect, split upward
create new root node

10. Definition of B-Tree
In general, a B-Tree of Order N has the following properties:
the root has at least two descendants, or is a leaf
each node has no mode that N descendents
each node that is not the root or a leaf has at least N/2 descendants
all leaf nodes are at the same level
a nonleaf node with k descendants contains k-1 key values

11. B+ Trees
Since search time = depth of tree, we need to keep the tree short and wide
Uneven tree (some full nodes and some near empty nodes, or leans to one side) creates poor performance
Using a slightly smarter split method during add keeps the tree short and balanced
B+ Tree is the de facto standard for databases

12. Storing a B-Tree in Files
Key
IRN
RRN P 4 1 W 8 2 -- 3 D 12 5 M 16 6 20 7 T 24 9 28 10 11 A 13 B 14 C 15 G 17 I 18 19 N 21 22 23 R 25 S 26 27
Data File
order does not matter
Index File
lists of indexes
each index: key, RRN
When inserting a new node, does its placement in the index file matter? P W D M P T W A B C D N P U W G I M R S T

13. Next Class Review of the entire semester Sorting and Binary Searching
FAT v. NTFS v. Linux
File Access Times
Fragmented Files
which is best storage method
Indexed
B-Tree
Hashed