1. Compsci 201 Trees and Tradeoffs
Owen Astrachan
Jeff Forbes
November 3, 2017
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Compsci 201, Fall 2017, Tree and Tradoffs

2. Compsci 201, Fall 2017, Tree and Tradoffs
S is for …
Stack
Last in, First Out, source of overflow!
Software
Joys and sorrows, eating the world
SQL
Structured Query Language
System
The .in’s and .out’s of Java
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Compsci 201, Fall 2017, Tree and Tradoffs

3. Compsci 201, Fall 2017, Tree and Tradoffs
Plan for the Day
Trees: Tradeoffs, In-Practice, In-Theory
Along the way we’ll revisit Stacks and Queues
Thinking Recursively
Developing recursive algorithms
Reasoning about recursive algorithms
Midterm and APT Quiz information
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Compsci 201, Fall 2017, Tree and Tradoffs

4. Why do we use search trees?
Fast insertion, removal, search: O(log n)
What is n?
When do trees “go bad” and what do we do?
Range queries are also important
All m values between low..high: O(m log n)
Can’t do that with a HashSet/HashMap
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Compsci 201, Fall 2017, Tree and Tradoffs

5. Compsci 201, Fall 2017, Tree and Tradoffs
Balanced Trees
In the 1960’s, Adelson-Velsky and Landis: AVL
O(log n) worst-case, because tree is rebalanced as needed
In 1978, Guibas and Sedgewick: Red-Black tree
Sedgewick continues to improve this!
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Compsci 201, Fall 2017, Tree and Tradoffs

6. What does insertion look like?
Simple recursive insertion into tree (accessed by root)
root = insert("foo", root);
TreeNode insert(TreeNode t, String s) {
if (t == null) t = new Tree(s,null,null);
else if (s.compareTo(t.info) <= 0)
t.left = insert(t.left,s);
else
t.right = insert(t.right,s);
return t; }
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Compsci 201, Fall 2017, Tree and Tradoffs

7. Notes on tree insert and search
In each recursive insert call
Tree parameter in call is either the left or right field of some node in the original tree
Will be assignment to a .left or .right field!
t.left = treeMethod(t.left,…)
Common idiom: modify and assign on return
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Compsci 201, Fall 2017, Tree and Tradoffs

8. Compsci 201, Fall 2017, Tree and Tradoffs
Insert and Removal
For insertion we can use iteration (see BSTSet)
Traverse left or right and “look ahead” to add
Removal has details, depends on # of children
Straightforward when zero or one child
Complicated when two children, find successor
See set code for complete cases
If right child, straightforward
Otherwise find node that’s left child of its parent (why?)
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Compsci 201, Fall 2017, Tree and Tradoffs

9. Compsci 201, Fall 2017, Tree and Tradoffs
Inorder w/o Recursion
public void inOrderStack(TreeNode root) {
Stack<TreeNode> stack = new Stack<>();
TreeNode current = root;
while (!stack.empty() || current != null) {
if (current != null) {
stack.push(current);
current = current.left; }
else {
current = stack.pop();
System.out.println(current.info);
current = current.right;
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Compsci 201, Fall 2017, Tree and Tradoffs

10. Aside: Wordladder Story
Ladder from ‘white’ to ‘house’
white, while, whale, shale, …
I can do that… optimally
My brother was an English major
My ladder is 16, his is 15, how?
There's a ladder that's 14 words!
The key is ‘sough’
Guarantee optimality!
QUEUE
I heard the puzzle on the way into Duke in the mid 90's. I called my brother and said I'd solve the problem. He said he would too. He used his brain, I wrote a program. Called him an hour alter and said "got it". He said "got it". I said my code was provably optimal and had a 16-word ladder. He said he had 15. WHAT?
I added "sough" to dictionary and got a 14-word ladder 
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Compsci 201, Fall 2017, Tree and Tradoffs

11. Queue for shortest path
public boolean findLadder(String[] words,
String first, String last){
Queue<String> qu = new LinkedList<>();
Set<String> set = new HashSet<>();
qu.add(first);
while (qu.size() > 0){
String current = qu.remove();
if (oneAway(current,last)) return true;
for(String s : words){
if (! set.contains(s) && oneAway(from,s)){
qu.add(s);
set.(s); } }
return false;
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Compsci 201, Fall 2017, Tree and Tradoffs

12. Shortest Path reprised
How does Queue ensure we find shortest path?
Where are words one away from first?
Where are words two away from first?
We use a set to avoid visiting the same word:
hot-dot-dog-dot-hot-hog-dog-dot …
What's path from white to house? We know there is one.
All words one-away from first are on Q and are added when first is dequeued/removed first time through loop. Each of these 1-away-from-start words comes off the queue and words that are one-away from these, or 2-away from start, are put on the Q. BUT it's a queue so all 2-away words go on after the 1-away.
In general, we remove N-away words before (N+1)-away words because of FIFO structure
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Compsci 201, Fall 2017, Tree and Tradoffs

13. Compsci 201, Fall 2017, Tree and Tradoffs
Shortest path proof
Just as we visited root before any trees in the LevelOrder Traversal
We visit all words one-away from start before all words two-away
We can combine queue with map to reconstruct the ladder
(Key,Value): value is word, key is what caused it to be put onto the queue
Don't dwell on this, but walk through it as a semi-formal argument, akin to a hand-wavy proof
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Compsci 201, Fall 2017, Tree and Tradoffs

14. Keeping track of ladder
Find w, a one-away word from current
Enqueue w if not seen
Call map.put(w,current)
Remember keys are unique!
Put word on queue once!
map.put("lot", "hot")
map.put("dot", "hot")
map.put("hat", "hot")
Remind students that we need to keep track of how words are connected to reconstruct the ladder. Keys in a map are unique, but we only put a word on the queue once! Why? Using a set. We can make this one time the time we make a word the key in a map. The value is what caused the word to be put onto the queue, see examples
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Compsci 201, Fall 2017, Tree and Tradoffs

15. Reconstructing Word Ladder
Run WordLaddersFull
See map and call to map.put(word,current)
What about when returning the ladder, why is the returned ladder in reverse order?
What do we know about code when statement adding (key,value) to map runs?
Run program with "white" "house" and with "voter" "crazy" and with "voter" "smart"
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Compsci 201, Fall 2017, Tree and Tradoffs

16. Compsci 201, Fall 2017, Tree and Tradoffs
BlobModel Revisited
Counting “blobs” using techniques from depth-first search and percolation
Rather than using recursion to visit all neighbors and find size of blob, use Queue of cells to manage the neighbors
Recursion is a stack-based model
This is a queue-based model
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Compsci 201, Fall 2017, Tree and Tradoffs

17. Mind your Stacks and Queues
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Compsci 201, Fall 2017, Tree and Tradoffs

18. Compsci 201, Fall 2017, Tree and Tradoffs
Jan Cuny
Program officer at National Science Foundation (NSF) Leading #CSforAll initiatives ABI Woman of Vision Award for Social Impact, 2016 Distinguished Educator Award
“All of toady’s kids will need – along with reading, writing, and arithmetic – a basic understanding of computation and the role it plays across a wide range of disciplines.”
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Compsci 201, Fall 2017, Tree and Tradoffs

19. Compsci 201, Fall 2017, Tree and Tradoffs
Midterm and APT Quiz
Practice midterm available
Focus of discussion section
Practice APT quiz available
Linked List question
Tree Question
Other question (linked list likely)
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Compsci 201, Fall 2017, Tree and Tradoffs

20. Compsci 201, Fall 2017, Tree and Tradoffs
Thinking about Trees
LeafSum
HeightLabel
LevelLabel
What to do in base case: null (and leaf node)
Combining results of recursive calls
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Compsci 201, Fall 2017, Tree and Tradoffs

21. Developing Recursive Ideas
When possible have the recursive calls mirror the structure of the problem: trees/lists are easy!
Store the result(s) of all call(s), use the results to return a value: call, store, compute, return
Sometimes a helper function with extra parameter is a good idea: height-label and level-label
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Compsci 201, Fall 2017, Tree and Tradoffs

22. Is Recursion Important?
Solving problems is important
Scaling solutions is important
Knowledge of tools that can be applied is important
The tool itself? ….
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Compsci 201, Fall 2017, Tree and Tradoffs

23. Time Changes Points of View
We argue that the notion of self reference should permeate first courses in computer science. If this is to be the case such courses should take a view far broader than “Wow, I can average 10 numbers with the skills I learned in my first programming course!”
Recursion is fundamental in computer science, whether understood as a mathematical concept, a programming technique, a way of expressing an algorithm, or a problem-solving approach. It is too important and too valuable to be belittled by showing a recursive factorial function in CS1, which conveys almost nothing of its power
Self Reference is an Illustrative Essential
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Compsci 201, Fall 2017, Tree and Tradoffs

24. Compsci 201, Fall 2017, Tree and Tradoffs
TIL
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Compsci 201, Fall 2017, Tree and Tradoffs

25. Leafsum Sum all the values in leaves of tree Base cases?
Recursive calls?
What should big-Oh be?
N-node tree?
Visit each node ….
Balanced or stringy …
Similar to tree height

26. Compsci 201, Fall 2017, Tree and Tradoffs
LeafSum correct?
What do we do with null tree? Why?
What value will always be returned? Why?
From base-case to combining recursive calls
Recurrence expression?
public class LeafSum {
public int sum(TreeNode t) {
if (t == null) return 0;
// something is missing here!
return sum(t.left) + sum(t.right); }
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Compsci 201, Fall 2017, Tree and Tradoffs

27. Compsci 201, Fall 2017, Tree and Tradoffs
LeafSum Now Correct!
What do we do with a leaf, why?
Why else isn’t needed, but ok
Why this is still O(n) for ALL TREES!
public class LeafSum {
public int sum(TreeNode t) {
if (t == null) return 0;
if (t.left == null &&
t.right == null) return 1;
return sum(t.left) + sum(t.right); }
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Compsci 201, Fall 2017, Tree and Tradoffs

28. Height Label: Call height
What is/are base cases?
Do we know how to determine height?
Create new nodes: info, left, right
What will recurrence be, similar to …
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Compsci 201, Fall 2017, Tree and Tradoffs

29. Compsci 201, Fall 2017, Tree and Tradoffs
Height and IsBalanced
int height(Tree root) { if (root == null) return 0; else { return 1 + Math.max(height(root.left), height(root.right)); } T(n) = 2T(n/2) + O(1) Visit both children Each visits both children, …
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Compsci 201, Fall 2017, Tree and Tradoffs

30. Compsci 201, Fall 2017, Tree and Tradoffs
Toward IsBalanced
boolean isBalanced(Tree root){
if (root == null) return true;
return
isBalanced(root.left) && isBalanced(root.right) &&
Math.abs(height(root.left)–height(root.right)) <= 1; }
T(n) = 2T(n/2) + O(n)
Visit both children
Each determines height!
3 gets height:4,1,7 then children!
4 gets height, 1,7 then children!
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Compsci 201, Fall 2017, Tree and Tradoffs

31. Compsci 201, Fall 2017, Tree and Tradoffs
Isomorphic trees
Two parameters, what are base cases?
If both s == null && t == null means ….
If one, not both that means …
sameShape(t.left,s.left) && …
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Compsci 201, Fall 2017, Tree and Tradoffs

32. Compsci 201, Fall 2017, Tree and Tradoffs
Height Label all Green
public TreeNode rewire(TreeNode t) {
if (t == null) return null;
return new TreeNode(height(t),
rewire(t.left),
rewire(t.right)); }
private int height(TreeNode t) {
if (t == null) return 0;
return 1 + Math.max(height(t.left),
height(t.right));
T(N) = 2T(N/2) + O(N) or .. T(N) = T(N-1) + O(N)
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Compsci 201, Fall 2017, Tree and Tradoffs

33. Compsci 201, Fall 2017, Tree and Tradoffs
YATP
Do two trees have the same shape?
If they are both empty/null they do
If left subtrees same? If right subtrees same?
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Compsci 201, Fall 2017, Tree and Tradoffs

34. Compsci 201, Fall 2017, Tree and Tradoffs
Quasi-Isomorphic
Base cases are similar: both null means …
If left same as right and right same as left?
If left same as left and right same as right?
How many recursive calls
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Compsci 201, Fall 2017, Tree and Tradoffs

35. What are all the calls for?
public String sameShape(TreeNode s, TreeNode t) {
if (s == null && t == null) return "same";
if (s == null || t == null) return "different";
if ("same".equals(sameShape(s.left, t.left)) &&
"same".equals(sameShape(s.right,t.right)))
return "same";
if ("same".equals(sameShape(s.left, t.right)) &&
"same".equals(sameShape(s.right, t.left)))
return "different"; }
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Compsci 201, Fall 2017, Tree and Tradoffs

36. Compsci 201, Fall 2017, Tree and Tradoffs
WOTO
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Compsci 201, Fall 2017, Tree and Tradoffs