CS3
Week 8: Recursion
Slide 1
Midterm 1
20
10
Std. Dev = 5.01
Mean = 21.8
N = 158.00
0
.0
29
.0
27
.0
25
.0
23
.0
21
.0
19
.0
17
.0
15
.0
13
.0
11
0
9.
0
7.
0
5.
• You did great
• If you need a regrade, see the person
that graded that
question
• Solutions are
available on the portal
(check
announcements).
SCALED
Slide 2
Question 1
• [Shotgun questions]
60
50
• (+ 3 x 5) may not
cause an error…
40
30
– (+ 3 'fred 5) will
20
10
Std. Dev = 1.19
Mean = 5.4
N = 158.00
0
3.0
4.0
5.0
6.0
7.0
Q1
Slide 3
Question 2
[add-em]
(add-em '(1 4 2 0 934 -3 5))
7
50
40
30
• The conditional was
tricky here:
– Needed to check for 3
things to get sent to
the base case
20
10
Std. Dev = 1.50
Mean = 5.5
N = 158.00
0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Q2
Slide 4
Question 3
[ranking cards]
• Scores are all over
the place (hallmark of
a bad question)
• Accessors!
40
30
20
10
Std. Dev = 2.43
Mean = 4.8
N = 158.00
0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Q3
Slide 5
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.
50
40
30
20
10
Std. Dev = 3.12
Mean = 6.7
N = 158.00
0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
Q4C
Slide 6
Question 5
[coins]
• Nice!
80
60
40
20
Std. Dev = 1.83
Mean = 7.7
N = 158.00
0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
Q5
Slide 7
Schedule
Oct 3
Midterm #1
Oct 10
Advanced recursion
Oct 17
Number-spelling Miniproject
Oct 24
Higher order procedures
Oct 31
More HOF
Lists! (instead of sentences and words)
Nov 7
Recursion (advanced)
Nov 14
Midterm #2
Slide 8
Problem: find all the even numbers in
sentence of numbers
(define (find-evens sent)
(cond ((empty?
(
sent)
;base case
'()
)
(
((odd?
(first sent)) ;rec case 1
1: odd
(find-evens (bf sent)) )
(
(else
;rec case 2: even
(se (first sent)
(find-evens (bf sent))) )
))
Slide 9
> (find-evens '(2 3 4 5 6))
sent = ( 2 3 4 5 6 )
(se 2
sent = ( 3 4 5 6 )
sent = ( 4 5 6 )
(se 4
sent = ( 5 6 )
sent = ( 6 )
(se 6
sent = ( )
()
 (se 2 (se 4 (se 6 ())
 (2 4 6)
Slide 10
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
Slide 11
Patterns in recursion
• Most recursions fall into a kind of patterns
– Students say that this helps them
• As a corollary, some recursions don’t!
Slide 12
• 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
Slide 13
• 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
Slide 14
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
Slide 15
Advanced recursion
columns (C)
0
1
2
3
4
5
...
0
1
r
o
w
s
1
1
1
2
1
2
1
3
1
3
3
1
(R)
4
1
4
6
4
1
5
1
5
10
10
5
1
...
...
...
...
...
...
...
...
...
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.
Slide 16
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)
Slide 17