1. Data Structures – An Introduction
Comp Sci Club
12 June 2014

2. After CS III AP… First of all – Congratulations on finishing CS III
There is much more to Comp Sci
(A lot more than I thought when I took the AP Exam!)

3. Overview (for Today’s Meeting)
Data Structures – some background
Binary Trees – BSTs, methods(), uses, etc.
*Types of Binary Trees
*Heaps, *HeapSort
Conclusion
* = new for CS Data Structures People

4. Data Structures Simple, efficient ways of organizing data
(Goes far Beyond the ‘Array’ and ‘ArrayList’ concepts taught in CS III!)

5. Data Structures
Abstract Data Type – more generalized forms of these structures (e.g. Queue, Stack, LinkedList, etc.)
Data Structure – how programmers/software developers implement these data types (ArrayQueue, LinkedStack/ArrayStack, Singly/DoublyLinkedList)
However…
There are some concepts which are more ambiguous, and can be considered part-“data type” and part-“data structure”
And, we will see a few of these in this presentation 

6. The Binary (Search) Tree
Organization of TreeNode objects: each ‘level’ of this structure has up to 2N Nodes.
Each Node has a reference to value; left, and right
Several different kinds of binary trees; a very common one is the Binary Search Tree
All Nodes left of root have VALUES less than the root; All Nodes right of root have VALUES greater than the root.
This principle applies to every Node, in fact.

7. The Binary (Search) Tree
BST represents hierarchy; allows very efficient searching or inserting; and can be manipulated easily J
Root Node → F M B H T
Leaves Y

8. Binary Trees – Specific Types*
Full Binary Tree – every node has zero or two ‘children’
Perfect Binary Tree – all leaf nodes lie on same level
Complete Binary Tree – every level is filled completely, except perhaps the last level (more detail later)

9. Infinite Binary Trees – A Digression
Stern-Brocot Tree – Mathematical model for the set of all positive rational numbers (known since the 1850s!)
Wikipedia: the cardinalities of these numbers
(the nodes themselves = א0; ways of permuting them = c)
There are numerous interesting facts about this model – see cut-the-knort.org and other sites for more info

10. Heaps

11. A Heap*
Disambiguation: I am not talking about THE Heap, which is considered the source for all dynamic data (i.e., how data is stored in an Object rather than a primitive type.) ….What do I mean then?

12. A Heap* Special Binary Tree – a complete binary tree
Bottom row (‘level’) fills from the left
Allows simple implementations (Array duality)
Max-heap property – root node is greater than all others; min-heap property – root node is less than all others
Technically, the heap can be seen as a data structure for a PriorityQueue
Queue, you may recall, is a built-in Java interface, modeling ‘FIFO’
PriorityQueue – more specific type that assigns a priority to each task/value, removing the highest Priority first

13. Heaps* Percolation Up/Down
Method of adding/deleting a Node to/from a Heap.
Involves swapping the last/final Node with the Root Node, then comparing the new ‘last’ with its parent Nodes to find where it should be.

14. HeapSort! Relatively efficient kind of sort
O (n log (n)) [for a worst case, this is not too bad]
Collects all elements into heap, sorts them inside the Heap, then remove each while updating the structure
In-place algorithm – uses small, constant memory
Somewhat complicated process
Make a heap out of the given info // Can be max or min!
Remove the root Node; replace it with the final Node in the heap
Perform comparisons (from left to right) to “rebalance” or reconfigure the heap again (method siftUp() or siftDown())
Repeat steps 2-3 until there are no Nodes remaining

15. Animated Simulations

16. Thank you Very Much!
This should be posted on within a week
Many thanks to all of you for a great year of CS Club
I exhort you to continue next year
(March competition!)
(Recruiting more people?)
Good luck on Final Exams; have a great summer 