1. CSE 214 – Computer Science II Binary Trees
Source:

2. Coding Exam 1 Next Friday, CS 2129 Linked Lists Only
2:15pm – 5:30 pm
Linked Lists Only
HWs 1 & 2
Sample exam has been posted to schedule page

3. Coding Exam 1 Review Session
Thursday evening
Room/Time TBD
I’ll post online/ class when I know

4. Trees A non-linear data structure
Components do not form a single simple sequence
Provide dramatic efficiencies for accessing data & problem reduction

5. Common Tree Terms Node – building block
Root – the top most node, provides access to all other nodes
Child node – a node accessible from another node, it’s parent node
Leaf – a node with no children (children are null)
BA $ 85.18
AIG $ 46.11
MO $ 72.53

6. Binary Trees Has a single root node
Each node stores a reference to up to 2 other different nodes
left child & right child
Each node has exactly 1 parent except the root
The root has no parents

7. More terms
Sibling – two nodes are siblings if they have the same parent
Ancestor – a node’s parent is its first ancestor, the parent of the parent is its next ancestor, etc.
Descendant – a node’ children are its first descendants, etc.
Subtree – Any node in a tree can also be viewed as the root of a new, smaller tree (can be called a subtree)
A binary tree has a left subtree & right subtree
Depth of a node – starting at a node, the number of steps up to reach the root
Depth of a tree – the maximum depth of any of its leaves

8. Full Binary Tree
Every leaf has the same depth and every non-leaf has 2 children
BA $ 85.18
AIG $ 46.11
MO $ 72.53
AA $ 35.72
AXP $ 45.11
MMM $ 79.95
T $ 37.88

9. Complete Binary Tree
Every level, except the deepest, must contain as many nodes as possible
The deepest level must have nodes packed as far left as possible
BA $ 85.18
AIG $ 46.11
MO $ 72.53
AA $ 35.72
AXP $ 45.11
MMM $ 79.95

10. Exercise Draw: a full binary tree with 15 nodes
a complete binary tree with 8 nodes

11. Full binary tree with 15 nodes

12. Complete binary tree with 8 nodes

13. Just to thoroughly confuse you
Many times complete binary trees are actually stored in packed arrays C O N F U S E D C O N F U S E D

14. Trees as Arrays No more node classes, just data in an array
Methods know how to access data properly
The root of the tree is always at index 0
For a node at index i, what is the index of its parent node?
floor((i-1)/2)
For a node at index I, what is the index of its child nodes?
left child at floor(2i + 1)
right child at floor(2i + 2)

15. Why would we do this? To squeeze performance out of our program
public class ArrayBinaryTree {
private int numElements = 15;
private Object[] data = new Object[numElements];
public ArrayBinaryTree() {} … }
To squeeze performance out of our program
more efficient memory management
faster for certain problems

16. Why not just use sequential arrays?
We may want the best of both worlds
efficient problem solving of trees
trees are good at reducing problems
trees are good at divide & conquer type algorithms
speed of processing of arrays
minimal memory overhead of arrays
we don’t need to store node data, like links between nodes
What’s a drawback?
if tree needs to add more nodes, it’s expensive
we may have to reconstruct array

17. But let’s start with something more familiar
Let’s look again at:
a tree managing class
a node class
Let’s have it store generic data

18. Whatever we want to put there, even other data structures
public class GenericCompleteBinaryTree <E> { private BinaryTreeNode root = null; public GenericCompleteBinaryTree() {} class BinaryTreeNode <E> protected E data = null; protected BinaryTreeNode<E> leftNode = null; protected BinaryTreeNode<E> rightNode = null; BinaryTreeNode( E initData, BinaryTreeNode<E> initLeftNode, BinaryTreeNode<E> initRightNode) data = initData; leftNode = initLeftNode; rightNode = initRightNode; }
What’s inside data?
Whatever we want to put there, even other data structures

19. Trees & Recursion To manipulate trees it’s common to:
keep the tree sorted
periodically rearrange the tree for better algorithmic payoff
have recursive methods that take a node or nodes as arguments
have recursive methods defined inside the node class
have tree methods call a method recursively on all nodes using one of the previously mentioned

20. More common uses of trees
Decision trees
AI trees
Dialog trees
Etc.