1 Recursion Overview l Introduction to recursion and recursive methods l Simple popular recursive algorithms l Writing recursive methods l Preview: 1-D array

2 Recursion l We saw earlier that a method could call another method and this could lead to a series of frames created on the stack. l In fact there is nothing preventing a method to call itself, or call another method which lead to calling the first method. Such a method is called a recursive method. l A well-defined recursive method has »A base case; »A recursive step which must always get “closer” to the base case from one invocation to another. l The code of a recursive method must be structured to handle both the base case and the recursive case. l Each call to the method sets up a new execution environment, with new parameters and local variables. l As always, when the method completes, control returns to the method that invoked it (which may be an earlier invocation of itself).

3 Recursion: The Factorial Function l The factocial function is defined as: n! = n*(n-1)*(n-2)*…*1 import java.io.*; class Factorial { public static int factorial(int num) { if (num == 0) return 1; else return num*factorial(num-1); } public static void main (String[] args) throws IOException { BufferedReader stdin= new BufferedReader( new InputStreamReader(System.in)); int n, answer; System.out.println("Enter an integer:"); n=Integer.parseInt(stdin.readLine()); answer = factorial(n); System.out.println("The factorial of " + n + " is " + answer); }

4 Recursion: Sample Execution Trace l Factorial: A Sample Trace factorial (6) = =(6*Factorial(5)) =(6*(5*Factorial(4))) =(6*(5*(4*Factorial(3)))) =(6*(5*(4*(3*Factorial(2))))) =(6*(5*(4*(3*(2*Factorial(1)))))) =(6*(5*(4*(3*(2*1))))) =(6*(5*(4*(3*2)))) =(6*(5*(4*6))) =(6*(5*24)) =(6*120) =720

5 Recursion: sumOfSquares l sumOfSquares: Specified as: sumSquares(m,n)=m^2+(m+1)^2+(m+2)^3+…+n^2 import java.io.*; class SumOfSquares { static int sumSquares(int from,int to) { if (from < to) return from*from + sumSquares(from+1,to); else return from*from; }

6 Recursion: sumOfSquares (cont.) l sumOfSquares (continued) public static void main (String[] args) throws IOException { BufferedReader stdin= new BufferedReader( new InputStreamReader(System.in);); int from, to, answer; System.out.println("Enter the smaller"); String input=stdin.readLine(); from = Integer.parseInt(input); System.out.println("Enter the bigger"); input=stdin.readLine(); to = Integer.parseInt(input); answer = sumSquares(from,to); System.out.println("Sum of Squares from "+ from + " to " +to+" is "+ answer); }

7 Recursion: sumOfSquares Execution Trace l sumSquares: A Sample Trace sumSquares (5,10) = =(25+sumSquares(6,10)) =(25+(36+sumSquares(7,10))) =(25+(36+(49+sumSquares(8,10)))) =(25+(36+(49+(64+sumSquares(9,10))))) =(25+(36+(49+(64+(81+(sumSquares(10,10))))))) =(25+(36+(49+(64+(81+100))))) =(25+(36+(49+(64+181)))) =(25+(36+(49+245))) =(25+(36+294)) =(25+330) =355

8 Recursion: The Fibonacci Function l The famous Fibonacci function is defined as fib 0 = 1 fib 1 = 1 fib n = fib(n-1)+fib(n-2), n>=2. import java.io.*; class Fibonacci { static int fibonacci (int num) { if (num == 0 || num == 1) return 1; else return (fibonacci(num-1) + fibonacci(num-2)); }

9 Recursion: The Fiboacci Function l Fibonacci public static void main (String[] args) throws IOException { BufferedReader stdin= new BufferedReader( new InputStreamReader(System.in)); int n, answer; System.out.println("Enter an integer:"); String input=stdin.readLine(); n = Integer.parseInt(input); answer = fibonacci(n); System.out.println("The "+n+ "th Fibonacci number is: "+ answer); }

10 Simple Recursive Algorithms (cont.) l Exercises: Write complete recursive programs for the following algorithms 1 power(x,y) that implements x^y using repeated additions and without using multiplication. Assume x to be a floating point value and y to be a nonnegative integer. 2 gcd(m,n) that implements the Euclid’s algorithm of finding the greatest common divisor of m and n. Assume m and n to be positive integers. 3. isPalindrome() which given a string prints an informative error message saying whether or not the given string is a palindrome (reads the same when read from left to right or from right to left).