Recursion

Circular Definition vs Recursive Definition Circular definition

oak A tree that grows from an acorn, which is a nut

produced by an oak tree

Recursion See recursion

Non-circular definitionRecursion

If you don’t get it, see recursion

Has a terminating case (i.e., “if you get it.”)

Recursive Definition

Recursive DefinitionUses the term being defined as part of the

definition Is not a circular definition

Factorial of n (n!)

n! = 1, when n = 0 // base casen! = n (n – 1)!, when n > 0

Recursive Function 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);

Recursive Factorial Function

fact(4) return 4 * fact(3)

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

Sum: 1 + 22 + 32 + 42 … + n2

Write a recursive function which returns the sum of squares of n integers.

Invocation

cin >> n;cout << sumSquares(n);

Sum: 1 + 22 + 32 + 42 … + n2

n = 1: sum = 1n = 2: sum = (1) + 22

n = 3 sum = (1 + 22)+ 32

n = 4: sum = (1 + 22 + 32) + 42

n = 5: sum = (1 + 22 + 32 + 42)+ 52

…n = n: sum = (1 + 22 + …(n -1)2)+ n2

Sum: 1 + 22 + 32 + 42 … + n2

long sumSquares(int n){ if (n == ) return 0; else return sumSquares(n – 1) + n * n; }

Recursive Sum of array elements

Given:

int list[] = {2, 4, 6, 8, 10};int count = 5;cout << sumArray(list, count);

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]

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]; }

Fibonacci Numbers

0 1 2 3 4 5 6 7 8 9 10 11 1 1 2 3 5 8 13 21 34 55 89 144

Fibonacci Numbers

fib(n) = 1, when n = 0fib(n) = 1, when n = 1fib(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);}

Golden Ratio

Greek and Renaissance Architect considered the golden ratio for a rectangle.

1

1.61803…

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 + ba a

a + 1 a 1

a b

= =φ=φ

=

Golden Ration & Fibonacci Nos.Fib Ratio Decimal 1 1/1 1.0 1 3/1 2.0 2 3/2 1.5 3 5/3 1.666. . . 5 8/5 1.6 8 13/8 1.625 13 21/13 1.615. . . 21 34/21 1.619. . . 34 55/34 1.617. . . . . . . . . . . . . . . . . . 1.6180 . . . φ

Your Turn

1. Write and test a recursive function which imports positive integer n and returns the sum of consecutive integers from 1 to n, inclusive.

2. Write and test a recursive function which returns xn.

3. Write a C++ program which generates the first n Fibonacci numbers, where n is input from the console.

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.