1. Trees 2
CSC 172
SPRING 2004
LECTURE 15

2. Why Structural Induction?
Q: Why do I want you to draw the pictures?
A: We know you can do the algebra. As you move on, the difficult part of the proofs is “setting up” the proof, from some situation. Sure, if someone sets it up for you, it’s just an “exercise” to solve. The real value is in being able to reason about a situation so as to set up a proof.

3. Process Real world Formal phenomena Rigorous Representation Knowledge
or artifact
Formal
Representation
“set up” of proof
Rigorous
Knowledge
Formal
techniques:
induction,
algebra
Informal
reasoning
abstraction
analysis

4. Kraft’s Inequality
Suppose that a binary tree has leaves {l1,l2,..lM} at depths {d1,d2,...,dM}, respectively.
Prove:

5. How do you set this up?

6. How do you set this up? Basis:
In a tree with zero nodes, the sum is zero
In a tree with one node
One leaf, depth 0

7. T1
∑1<=1 T2
∑2<=1
Same number of total leaves, all depths increase by 1
So, in new Tree ∑1<=1/2 && ∑2<=1/2

8. A tree viewed recursively

9. Size of a tree
Recursive view
ST = SL + SR + 1

10. Size of a Tree public static int size (BinaryNode t) { if (t == null)
return 0 ;
else
return 1 + size(t.left) + size(t.right); }

11. Height of a tree Recursive view of height calculation
HT = Max(HL,HR) + 1;

12. Height of a Tree public static int height (BinaryNode t) {
if (t == null) return -1 ;
else
return
1 + Math.max(height(t.left))
,(height(t.right)); }

13. Tree Traversal
Preorder
Postorder
Inorder

14. Preorder public void printPreOrder() { System.out.println(element);
if (left != null)
left.printPreOrder();
if (right != null)
right.printPreOrder(); }

15. Postorder public void printPostOrder() { if (left != null)
left.printPostOrder();
if (right != null)
right.printPostOrder();
System.out.println(element); }

16. Inorder public void printInOrder() { if (left != null)
left.printInOrder(); System.out.println(element);
if (right != null)
right.printInOrder(); }

17. Tree Iterators Stack use to implement recursion Linear
Good demonstration of class hierarchy in design

18. TreeIterator class abstract class TreeIterator {
public TreeIterator( BinaryTree theTree ) { t = theTree; current = null; }
abstract public void first( );
final public boolean isValid( ) { return current != null;}
final public Object retrieve( ) {
if( current == null ) throw new NoSuchElementException( "TreeIterator retrieve" );
return current.getElement( ); }
abstract public void advance( );
protected BinaryTree t; // Tree
protected BinaryNode current; // Current position

19. Stacks for Recursion We have seen recursion
We know we can use recursion
But how do the virtual machines actually *do* recursion
We can use a stack to implement recursion
Lets consider the PostOrder traversal implementing recursion in the iterator with a stack

20. The Algorithm The top of the stack is the “current” node
We can be in one of 3 states while at the current node
About to recurse to the left
About to recurse to the right
About to process the current node
So, we put each node on the stack 3 times
The 3rd time we pop, we process

21. Stack states during postorder

22. Implementation class PostOrder extends TreeIterator {
public PostOrder( BinaryTree theTree ) {
super( theTree );
s = new ArrayStack( );
s.push( new StNode( t.getRoot( ) ) ); }
public void first( ) {
s.makeEmpty( );
if( t.getRoot( ) != null ) {
advance( );

23. Implementation protected static class StNode { BinaryNode node;
int timesPopped;
StNode( BinaryNode n ){
node = n;
timesPopped = 0; }
/** An internal stack if visited nodes. */
protected Stack s;

24. Implementation: advance
public void advance( ) {
if( s.isEmpty( ) ){ // check for null
if( current == null )
throw new NoSuchElementException(
"PostOrder Advance" );
current = null;
return; }

25. Implementation StNode cnode; for( ; ; ) {
cnode = ( StNode ) s.topAndPop( );
if( ++cnode.timesPopped == 3 ) { current = cnode.node; return; }
s.push( cnode );
if( cnode.timesPopped == 1 ) {
if( cnode.node.getLeft( ) != null ) s.push( new StNode( cnode.node.getLeft( ) ) ); }
else { // cnode.timesPopped == 2
if( cnode.node.getRight( ) != null ) s.push( new StNode( cnode.node.getRight( ) ) );