1. CSE 214 – Computer Science II B-Trees
Source:

2. Coding Exam 3 Friday, 12/4 – Starting at 2:15 Topics:
Stacks (using linked lists & arrays)
Queues (using linked lists & arrays)
Hash Tables (open address & chained)
Heaps
Sample exam is posted to schedule page
note, past exams have not covered identical topics
sample exam only has stacks & queues

3. Exam Review Session
Thursday, 8 – 9:30 pm
CS 2129

4. We’ve studied many types of trees already
Binary Search Trees
K-ary Trees
Complete Trees
Full Trees
Octrees

5. There are many, many, more
B-Trees
Red-Black trees
2-3 Trees
Etc.
There are custom trees for solving many problems

6. Let’s examine on such tree: B-Trees
There will be final exam questions on B-Trees
conceptual only
Note: this is perhaps the toughest topic we’ll cover this semester
Why is it tough?
complexity of implementation
at first glance, benefits aren’t obvious
Why are we covering it?
they are an important technology (DBMSs love them)

7. B-Tree applications
Foundation for database and file system data management
Why are they used?
O(log N) amortized time for accessing, insertion, deletion
What’s amortized time?
average time per calculation measure over a large number of operations

8. A B-Tree with Letters for data
What characteristics do you notice about this B-Tree?
it’s sorted, each node has multiple data points, nodes may have many children, the number of children is related to the amount of data a node has, all leaves are on the same level
Ref:

9. The 6 B-Tree Rules
The root can have as few as one element (or no elements if it has no children). Every other node has at least MINIMUM elements.
The MAXIMUM number of elements in a node is twice the value of MINIMUM.
The elements of each B-Tree node are stored in a partially filled array, sorted from the smallest element (at index 0) to the largest element (at the final used position of the array).
The number of sub-trees below a non-leaf node is always one more than the number of elements in the node
For any non-leaf node: (a) An element at index i is greater than the elements in sub-tree number i of the node, and (b) an element at index i is less than all the elements in sub-tree number i + 1 of the node.
Every leaf in a B-tree has the same depth

10. A note about MAXIMUM & MINIMUM
These values are selected through tuning
What’s tuning?
examining performance at runtime
making appropriate changes to optimize
MINIMUM selected based on:
amount of data for the B-Tree
likely operations to perform on tree
may be in hundreds or thousands in a database
Selecting MINIMUM affects:
size of nodes
depth of tree
efficiency of various operations

11. The 6 B-Tree Rules
The root can have as few as one element (or no elements if it has no children). Every other node has at least MINIMUM elements.

12. The 6 B-Tree Rules
2. The maximum number of elements in a node is twice the value of MINIMUM.

13. The 6 B-Tree Rules
3. The elements of each B-Tree node are stored in a partially filled array, sorted from the smallest element (at index 0) to the largest element (at the final used position of the array).

14. The 6 B-Tree Rules
4. The number of sub-trees below a non-leaf node is always one more than the number of elements in the node

15. The 6 B-Tree Rules
5. For any non-leaf node: (a) An element at index i is greater than all the elements in sub-tree number i of the node, and (b) an element at index i is less than all the elements in sub-tree number i + 1 of the node.

16. The 6 B-Tree Rules
6. Every leaf in a B-tree has the same depth

17. What would each node need?
an array of data
counter for data
array of child nodes
counter for children
What should our B-Tree’s data be?
any sortable object
we can specify our own comparison criteria