1. Recursion DRILL: Please take out your notes on Recursion
double Power (int base, int exp) {
double ans = 1;
for (int j = 0; j < exp; j++)
ans = ans * base;
return ans; }
Procedures:
1. Go over Sorting and Searching Practice
2. Notes on Recursion
3. Recursion Practice
4. Recursion Programs
HW : Recursion Ditto
Start working on Recursion Problems

2. Objective:
You will solve programming problems that contain repetition by defining, analyzing, and completing activities which involve recursion

3. Recursion
In Java, any method can call another method. A method can even call When a method calls itself, it is making a
recursion : the process of a method calling itself
Recursion is a powerful technique that can be used in place of
Any recursive method can be rewritten using iteration.
itself
recursive call
iteration (looping).

4. To calculate the result of raising an integer to a positive power, if x is an integer and n is a positive integer, the formula xn is :
xn = x * x * x * x * * x
n times
We could also write this formula as :
xn = x * (x * x * x * * x)
times
or as
this is a
n - 1
xn = x * xn - 1
recursive definition

5. recursive definition : a definition in which something is defined
in terms of smaller versions of itself.
256= 25 * 25 * 25 * 25* 2 5* 25
= 25 * (25 * 25 * 25 * 25 * 25) = 25 * 255
= 25 * (25 *(25 * 25 * 25 * 25)) = 25 * (25 * 254)
= 25 * (25 *(25 * (25 * 25 * 25))) = 25 * (25 * (25 * 253))
= 25 * (25 *(25 * (25 *( 25 *25)))) = 25 * (25 * (25 * (25 * 252)))
= 25 * (25 * (25 * (25 *( 25 *(25))))) = 25 * (25 * (25 * (25 * (25 * (251)))

6. All recursive functions must have a
All recursive functions must have a The base case the chain of recursive calls.
base case (stopping state): a case that does generate a recursive call.
What is the base case in the problem of 256?
base case
terminates
not
250

7. Write Power Recursively
int Power (int base, int exp) { }
if ( exp = = 0) // base case
return 1;
else // recursive call
return base * Power(base, exp - 1)

8. How can I figure out what is returned from the following call?
Power (25, 6)

9. 5 * (4*(3*(2*1!)))= 5*(4*(3*(2*(1*0!))))
You Try :
1. Express 5! in terms of a smaller definition of itself.
2. What is the base case for Factorial?
3. Write a method called Factorial that uses recursion to determine the factorial of a number.
5*(4!) = 5 *(4*3!) = 5*(4*(3*2!)) =
5 * (4*(3*(2*1!)))= 5*(4*(3*(2*(1*0!))))
0! n = 0

10. int Factorial (int n) {
if (n == 0)
return 1;
else
return n * Factorial (n - 1); }

11. ` What is Happening in Memory
When a function is called (either recursively or non-recursively), the computer system creates temporary storage for the actual parameters and the methods’s local variables in a region of memory called the run-time stack. When the function returns, its parameters and local variables are released from the run-time stack.
stack : a dynamic data structure where access can only be made from___________.
A stack is a ________ structure - __________________ `
one end
LIFO
Last In First Out

12. Each call to a function can be thought of as adding a “tray” to the run-time stack.
Suppose there is a call to Factorial in main.
ans = Factorial (5);

13. 1 1 1 2 2 3 6 4 24 5 120 int Factorial (int n) { if (n == 0) return 1;
else
return n * Factorial (n - 1); } 1 1 1 2 2 3 6 4 24 5
120

14. There is only a finite amount of space on the run-time stack
There is only a finite amount of space on the run-time stack. If the function never returns a value, eventually all the memory space on the stack is used and the program crashes with an error message such as “ ”.
A function returns when there is a or when a void function reaches the
run-time stack overflow
return statement
closing brace

15. int Recur (int num) {
if (num ==0)
return 0;
else
return num + Recur(num-1); }
What is returned from the call Recur (4)?

16. return num + Recur(num-1); }
int Recur (int num) {
if (num ==0)
return 0;
else
return num + Recur(num-1); }
What is returned from the call Recur (-4)?
Infinite Recursion: recursive calls that do not stop.
Infinite Recursion! You will eventually run out of memory

17. Write recursive functions to print a string forwards and backwards.

18. Tail Recursion: a recursive algorithm in
// Recursive Method to print a string
void Print (String word, int n) { }
System.out.print( word.charAt(n));
if (n != word.length()-1)
Print (word, n+1);
Tail Recursion: a recursive algorithm in
which no statements are executed after the
return from the recursive call.

19. // Recursive Method to print a string backwards
void PrintBack (apstring word, int n) { }
System.out.print( word.charAt(n));
if (n != 0)
PrintBack (word, n-1);

20. Recursion : Wrap-up!
takes more ___________________ and ___________________
reduces the ___________________ of a problem - recursive solutions are usually simpler solutions to a problem
must have a ___________________
a recursive algorithm can be implemented nonrecursively by using _________________
nonrecursive solutions are often more efficient in terms of time and space, but are usually less ___________________.
time
space
complexity
base case
iteration
elegant

21. Class Exercise

22. return Combo (amount, value – 1) + Combo (amount – value, value);
int Combo (int amount, int value) {
if ((amount < 0) || (value == 0))
return 0;
else
if (amount == 0)
return 1;
return Combo (amount, value – 1) + Combo (amount – value, value); }
int main ()
cout << Combo (4, 3);
What is output by the program?
Work it out!

23. Hints! Power – you already know
PowerOfTwo – think num%2 and num/2 and test for remainder
Twos - very similar to PowerOf Two except leave a count of 1 or 0
Binary – think n%2 and n/2