1. Slide 1 CS3 Week 8: Recursion

2. Slide 2 Midterm 1 You did great If you need a re- grade, see the person that graded that question Solutions are available on the portal (check announcements).

3. Slide 3 Question 1 [Shotgun questions] (+ 3 x 5) may not cause an error… –(+ 3 'fred 5) will

4. Slide 4 Question 2 [add-em] ( add-em '(1 4 2 0 934 -3 5))  7 The conditional was tricky here: –Needed to check for 3 things to get sent to the base case

5. Slide 5 Question 3 [ranking cards] Scores are all over the place (hallmark of a bad question) Accessors!

6. Slide 6 Question 4 [day-span] Question 4c – the short answers – was a good one (I think). Some had trouble deciding between –conditional, –the base case, –making the problem smaller, –calling the function recursively –combining the recursive calls.

7. Slide 7 Question 5 [coins] Nice!

8. Slide 8 Schedule Oct 3Midterm #1 Oct 10Advanced recursion Oct 17Number-spelling Miniproject Oct 24Higher order procedures Oct 31More HOF Lists! (instead of sentences and words) Nov 7Recursion (advanced) Nov 14Midterm #2

9. Slide 9 Problem: find all the even numbers in sentence of numbers (define (find-evens sent) (cond ( ;base case ) ( ;rec case 1: odd ) ( ;rec case 2: even ) )) (define (find-evens sent) (cond ((empty? sent) ;base case '() ) ((odd? (first sent)) ;rec case 1 (find-evens (bf sent)) ) (else ;rec case 2: even (se (first sent) (find-evens (bf sent))) ) ))

10. Slide 10 > (find-evens '(2 3 4 5 6))  (se 2 (se 4 (se 6 ())  (2 4 6) (se 2 (se 4 (se 6 () sent = ( 2 3 4 5 6 ) sent = ( 3 4 5 6 ) sent = ( 4 5 6 ) sent = ( 5 6 ) sent = ( 6 ) sent = ( )

11. Slide 11 Why is recursion hard? ONE function: –replicates itself, –knows how to stop, –knows how to combine the “replications” There are many ways to think about recursion: you absolutely do not need to understand all of them. –"down-up": recursion as an extension of writing many specific functions –"many base cases": recursion as using a clone, once you have many base cases

12. Slide 12 Patterns in recursion Most recursions fall into a kind of patterns –Students say that this helps them As a corollary, some recursions don’t!

13. Slide 13 Mapping –does something to every part of the input sentence –E.g., square-all Counting –Counts the number of elements that satisfy a predicate –E.g., count-vowels, count-evens Finding –Return the first element that satisfies predicate (or, return rest of sentence) –E.g., member, member-even

14. Slide 14 Filtering –Keep or discard elements of input sentence –E.g., keep-evens Testing –A predicate that checks that every or any element of input satistfies a test –E.g., all-even? Combining –Combines the elements in some way… –E.g., sentence-sum

15. Slide 15 What recursions aren’t covered by these patterns? Weird ones like reverse, or downup –... bowling... "Advanced" recursions: –when it does more than one thing at a time –Ones that don’t traverse a single sentence E.g., mad-libs takes a sentence of replacement words [e.g., ‘(fat Henry three) ] and a sentence to mutate [e.g., ‘(I saw a * horse named * with * legs) ] –Tree recursion: multiple recursive calls in a single recursive step

16. Slide 16 Advanced recursion columns (C) r o w s (R) 012345... 01 111 2121 31331 414641 51510 51... Pascal’s Triangle How many ways can you choose C things from R choices? Coefficients of the (x+y)^R: look in row R etc.

17. Slide 17 pair-all Write pair-all, which takes a sentence of prefixes and a sentence of suffixes and returns a sentence pairing all prefixes to all suffixes. –(pair-all ‘(a b c) ‘(1 2 3))  (a1 b1 c1 a2 b2 c2 a3 b3 c3) –(pair-all ‘(spr s k) ‘(ite at ing ong))  (sprite sprat spring sprong site sat sing song kite kat king kong)