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Chapter 11

Recursion

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Recursive Methodso A recursive method is a method that calls itself directly

or indirectly.o A recursive method has two major steps:

1. recursive step in which the method calls itself2.base step which specifies a case with a known

solution

o The method should select one of two steps based on a criteria:

Example:recursive step: fact(n) = n * fact(n-1)base step: fact(0) = 1

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Constructing Recursion

o To construct a recursive algorithm you have to find out:

1.Recursive step

2.Base step

o A selection structure is then used to determine which

step to take.

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General Algorithm

if (stopping condition) then

solve simple problem (base)

else

use recursion to solve smaller problem

combine solutions from smaller problem

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Recursive Methods0! = 1 (By Definition!)n! = n x (n – 1) ! If n > 03! = 3 x 2!2! = 2 x 1!1! = 1 x 0!

0! = 1 (Base Case!)

1! = 1 x 0! = 1 x 1 = 12! = 2 x 1! = 2 x 1 = 23! = 3 x 2! = 3 x 2 = 6

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Recursive Methods• recursive step: fact(n) = n * fact(n-1)• base step: fact(0) = 1

fact(4) = 4 * fact(3) = 4 * (3 * fact(2)) = 4 * (3 * (2 * fact(1)))= 4 * (3 * (2 * (1 * fact(0)))) = 4 * (3 * (2 * (1 * 1))) = 4 * (3 * (2 * 1)) = 4 * (3 * 2) = 4 * 6 = 24

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Recursive Factorial Method

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Recursive Factorial Method

public static int fact(int num){ if (num = = 0) return 1; else return num * fact(num – 1);}

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Fibonacci numbers

n 0 1 2 3 4 5 6 7 8 9 10 11Fib(n) 0 1 1 2 3 5 8 13 21 34 55 89

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public int fib(int n) {    if(n <= 1) {        return n;    } else {        return fib(n - 1) + fib(n - 2);    }}

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Convert from decimal to binary

This method converts an integer number to its binary equivalent.

Base step: dec2bin(n) = n if n is 0 or 1

Recursive step: dec2bin(n) = dec2bin (n/2) , (n mod 2)

Algorithm dec2bin(n):

If n < 2 Print nelse

dec2bin(n / 2) Print n mod 2

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Example (Recursion) class Method { public static void dec2bin( int n){

if ( n < 2 ) System.out.print( n );

else { dec2bin( n / 2 ); System.out.print( n % 2 ); } }}class Dec2Bin{

public static void main(String [] arg){ int i=10;

dec2bin(i);}

}

Output:1010

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Example (iterative)

public static void dec2bin(int n){ String binary =""; while ( n >= 1 ) { binary = n%2 + binary;

n /= 2; }

System.out.print(binary);}

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Recursion or Iteration?

Two ways to solve particular problem: Iteration Recursion

Iterative control structures use looping to repeat a set of statements.

Tradeoffs between two options: Sometimes recursive solution is easier. Recursive solution is often slower.

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Exercise

1. Write a recursive method to find the greatest common

divisor (GCD) of two integer n and m.

2. Write a recursive method to find Xn given the double X

and the integer n.

3. Consider a Boolean array b filled with Boolean values.

Write a recursive method boolean allTrue() that returns

true if all values are true and returns false otherwise.