1. COMP 103
RECURSION

2. RECAP-TODAY RECAP TODAY Recursion Selection Sort, Insertion Sort2
RECAP
Selection Sort, Insertion Sort
Cost Analysis – those were the “slow” sorts: O(n2)
TODAY
Recursion

3. Recursion Methods can call other methods....3
Methods can call other methods....
...so could a method call itself ?
Something is recursive if it is defined in terms of itself
We will look at:
tail recursion (the simplest)
diagrams of what's going on
recursion with a wrapper method
nested recursion
multiple recursion
returning values

4. Example 4
/** Draw stream of bubbles from (x,y) to the top of the screen */
public void drawBubbles(int x, int y){
if ( y>0 ) {
drawOneBubble(x, y);
drawBubbles(x, y-15); }
y=0
y=70
x=30
drawBubbles (30, 40);
drawOneBubble(30,40);
drawBubbles (30, 55);
drawOneBubble(30,55);
drawBubbles (30, 70);
drawOneBubble(30,70);

5. Tracing the calls drawBubbles(30, 70) drawBubbles(30, 55)
public void drawBubbles(int x, int y){
if ( y>0 ) {
drawOneBubble(x, y);
drawBubbles(x, y-15); }
return
drawBubbles(30, 70)
drawOneBubble(30, 70)
drawBubbles(30, 55)
return
drawOneBubble(30, 55)
drawBubbles(30, 40)
return
drawOneBubble(30, 40)
drawBubbles(30, 25)
return
drawOneBubble(30, 25)
drawBubbles(30, 10)
return
drawOneBubble(30, 10)
drawBubbles(30, -5)
return

6. Recursion with wrapper method6
/** Draw stream of bubbles from (x,y) to the top of the screen
the bubbles should get larger as they go up. */
public void drawBubbles(int x, int y){
UI.setColor(Color.blue);
drawBubblesRec(x, y, 10); }
/** Recursive method */
public void drawBubblesRec(int x, int y, int size){
if ( y>0 ) {
UI.fillOval(x, y, size, size);
drawBubblesRec(x, y-size-6, size+2);
Initial size
Condition
Do First
Do the rest
Arguments for the rest

7. Tail recursion vs nested recursion.7
Tail recursion:
recursive call is the last step in the method.
easier to understand because don’t have to “go back” after call.
Nested recursion:
recursive call is in the middle.
have to go back to previous call to finish it off.
Example: Print an “onion” : ( ( ( ( ( ( ( ) ) ) ) ) ) )
public void onion (int layers){
UI.print( “(“ );
if (layers > 1) onion(layers-1);
UI.print( “)“ ); }
open
do the inside
close

8. Nested Recursion at work8
onion(4) ⇒
public void onion (int layers){
UI.print( “(“ );
if (layers > 1) onion(layers-1);
UI.print( “)“ ); ( ( ( ( ) ) ) )
public void onion (int layers){
UI.print( “(“ );
if (layers > 1) onion(layers-1);
UI.print( “)“ ); }
4 .
public void onion (int layers){
UI.print( “(“ );
if (layers > 1) onion(layers-1);
UI.print( “)“ ); }
3 . ✔
public void onion (int layers){
UI.print( “(“ );
if (layers > 1) onion(layers-1);
UI.print( “)“ ); }
2 . ✔ ✔
public void onion (int layers){
UI.print( “(“ );
if (layers > 1) onion(layers-1);
UI.print( “)“ ); }
1 . ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔ ✔

9. Alternative Visualisation9
onion(4) ✔
print( “(” )
print( “)” )
onion(3) ✔ ✔ ✔
print( “(” )
print( “)” )
onion(2) ✔ ✔ ✔
print( “(” )
print( “)” )
onion(1) ✔ ✔ ✔
print( “(” )
print( “)” ) ✔ ✔
To understand recursive methods: draw a diagram!

10. Kinds of Recursion Tail recursion vs. Nested recursion10
Tail recursion vs. Nested recursion
Tail: recursive call is the last step in a method
⇒ don’t have to “go back” and do more.
Nested: recursive call is in the middle
⇒ action … recursive call … more actions
Single recursion vs. Multiple recursion
Single: method only makes one recursive call
Multiple: method makes more than one recursive call
Single
Recursion

11. Multiple Recursion Draw a recursive arch-wall:11
Draw a recursive arch-wall:
Consists of an arch with two half size arch-walls on top of it.

12. Multiple Recursion to draw an ArchWall of given base size (wd,ht):12
to draw an ArchWall of given base size (wd,ht):
draw a base arch of size (wd,ht)
if wd is not too small
draw a half size archwall on the left
draw a half size archwall on the right
public void archWall (int x, int y, int wd, int ht){
drawArch(x, y, wd, ht);
if ( wd > 20 ) {
int w = wd/2; // width of next smaller arch
int h = ht/2; // height of next smaller arch
archWall(x, y-h, w, h); // left half
archWall(x+w, y-h, w, h); // right half }

13. Tracing the execution:13
archWall(10, 300, 80, 40)
aw(50,280,40,20)
drawArch
aw(10, 280, 40, 20)
drawArch
aw(10,270,20,10)
aw(30,270,20,10)
drawArch
aw(50,270,20,10)
aw(70,270,20,10)
drawArch
drawArch
drawArch
drawArch

14. Recursion that returns a value14
What if the method returns a value?
⇒ get value from recursive call, then do something with it
typically, perform computation on value, then return answer.
Compound interest
value at end of n th year =
value at end of previous year * (1 + interest).
value(deposit, year) = deposit [if year is 0]
= value(deposit, year-1) * (1+rate)

15. Recursion returning a value15
/** Compute compound interest of a deposit */
public double value(double deposit, double rate, int year){
if (year == 0)
return deposit;
else {
int prev = value(deposit, rate, year-1);
return prev * (1 + rate); }

16. Recursion with return: execution16
public double value(double deposit, double rate, int year){
if (year == 0) return deposit;
else {
int prev = value(deposit, rate, year-1); ← step 1
return prev * (1 + rate); ← step 2 }
value(1000, 0.1, 4)
value(1000, 0.1, 3) * 1.1
value(1000, 0.1, 2) * 1.1
value(1000, 0.1, 1) * 1.1
value(1000, 0.1, 0) * 1.1
1464.1
1331
1210
1100
1000