1. Recursion

2. Math Review Given the following sequence: a 1 = 1 a n = 2*a n-1 OR a n+1 = 2*a n What are the values of the following? a 2 = a 3 = a 4 =

3. What is recursion? circular definition a function call within a function definition “using a function inside itself” Example: a(1) = 1 a(n) = 2*a(n-1) int a(int n) { if (n == 1) return 1; return 2*a(n-1); }

4. The Base Case Every recursive algorithm has a starting/ending  base case What was the base case for the previous example? What would happen if we did not have the base case?

5. Example What is the base case and equation for the following sequence? a 0 = 1 a 1 = 1 a 2 = 2 a 3 = 6 a 4 = 24 a 5 = 120 a 6 = 720

6. Recursive Function Construction 1. Always start with checking the base case 2. Recursively call (use) the function Don’t forget to include any additional calculations

7. Example a 0 = 1 a n = n * a n-1 OR a(0) = 1 a(n) = n * a(n-1) OR factorial(0) = 1 factorial(n) = n * factorial (n-1) int factorial(int n) { if (n == 0) return 1; return n * factorial(n-1); }

8. Tracing a Recursive Function public static void main( String[] args ) { System.out.println( factorial(4) ); } public int factorial(n) { if (n == 0) return 1; return n * factorial(n-1); }

9. Your Turn What is the base case and equation for the following sequence? a 0 = 1 a 1 = 1 a 2 = 2 a 3 = 3 a 4 = 5 a 5 = 8 a 6 = 13

10. Your Turn Implement the recursive function for the fibonacci sequence

11. Fibonacci int fib(int n) { if (n == 0) return 1; if (n == 1) return 1; return fib(n-1) + fib(n-2); }

12. Recursion Advantages elegant solutions often fewer variables and fewer lines of code (LOCs) many problems are best solved using recursion (e.g. factorial, fibonacci)

13. Recursion Disadvantages recursive overhead  slower than iteration tricky to follow