1. Recursion

2. Circular Definition vs Recursive Definition
oak
A tree that grows from an acorn, which is a nut produced by an oak tree
Recursion
See recursion

3. Non-circular definition
Recursion
If you don’t get it, see recursion
Has a terminating case (i.e., “if you get it.”)

4. Recursive Definition Recursive Definition
Uses the term being defined as part of the definition
Is not a circular definition
Factorial of n (n!) n! = 1, when n = // base case n! = n (n – 1)!, when n > 0

5. Recursive Function Invokes itself within the function
Is an example of divide-and-conquor technique
Recursive factorial function int fact(int n){ if (n == 0) return 1; else return n * fact(n – 1); }
Invocation cout << fact(4);

6. Recursive Factorial Function
fact(4) return 4 * fact(3)
Recursive Factorial Function
fact(3) return 3 * fact(2)
fact(2) return 2 * fact(1)
fact(1) return 1 * fact(0)
fact(0) return 1
fact(1) return 1 * 1
fact(2) return 2 * 1
fact(3) return 3 * 2
fact(4) return 4 * 3

7. Sum: … + n2
Write a recursive function which returns the sum of squares of n integers.
Invocation cin >> n; cout << sumSquares(n);

8. Sum: … + n2
n = 1: sum = 1 n = 2: sum = (1) + 22 n = 3 sum = (1 + 22)+ 32 n = 4: sum = ( ) + 42 n = 5: sum = ( )+ 52 … n = n: sum = ( …(n -1)2)+ n2

9. Sum: … + n2
long sumSquares(int n){ if (n == ) return 0; else return sumSquares(n – 1) + n * n; }

10. Recursive Sum of array elements
Given: int list[] = {2, 4, 6, 8, 10}; int count = 5; cout << sumArray(list, count);

11. Recursive Sum of array elements
n = 1: sum = list[0] n = 2: sum = (list[0]) + list[1] n = 3: sum = (list[0] + list[1]) list[2] n = 4: sum = (list[0] + list[1] + list[2]) list[3] n = n: sum = (list[0] + list[1]… list[n – 2]) list[n – 1]

12. Recursive Sum of array elements
long sumArray(int list[], int n){ if (n == 1) return list[0]; else return sumArray(list, n – 2) list[n – 1]; }

13. Fibonacci Numbers

14. Fibonacci Numbers
fib(n) = 1, when n = 0 fib(n) = 1, when n = 1 fib(n) = fib(n -1) + fib(n – 2), when n > 1
Fibonacci Function int fib(int n){ if (n == 0 || n == 1) return 1; else return fib(n – 1) + fib(n – 2); }

15. Golden Ratio
Greek and Renaissance Architect considered the golden ratio for a rectangle. 1 …

16. Golden Ratio
Euclid: "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” (Elements) a b
a + b a b
a + 1 a 1 φ φ = = = = a a

17. Golden Ration & Fibonacci Nos.
Fib Ratio Decimal / / / / / / / / /
 φ

18. Your Turn
Write and test a recursive function which imports positive integer n and returns the sum of consecutive integers from 1 to n, inclusive.
Write and test a recursive function which returns xn.
Write a C++ program which generates the first n Fibonacci numbers, where n is input from the console.

19. Your Turn
Write and test a recursive function which imports an int array, start index, & last index and prints the list contents from position start to last.
Write and test a recursive function which imports an int array and its count and prints the list contents in reverse order.