1. 1 Object-Oriented Programming Using C++ CLASS 2

2. 2 Linear Recursion Summing the Elements of an Array Recursively Algorithm LinearSum(A, n): Input: An integer array A and int n Output: The sum of the first n int in A if n = 1 then return A[0] else return LinearSum(A, n-1) + A[n-1]

3. 3 Linear Recursion Reverse an Array by Recursion Algorithm ReverseArray(A, i, n): Input: An integer array A and int i, n Output: The reversal of n integers in A starting at index i if n > 1 then Swap A[i] and A[i+n-1] Call ReverseArray(A, i+1, n-2) return

4. 4 Linear Recursion Computing Powers Algorithm Power(x, n): Input: The number x and int n >= 0 Output: The value x power n if n = 0 then return 1 if n is odd then y  Power(x, (n-1)/2) return x * y * y else y  Power(x, n/2) return y * y

5. 5 Tail recursion Using recursion can often be a useful tool for designing algorithms, but this usefulness does come at a modest cost Whenever we use recursion function to solve a problem, we have to use some of the memory locations in our computer to keep track of the state of each active recursive call We can use the stack data structure to convert a recursive algorithm into a non-recursive algorithm, easily the ones which use tail recursion A tail recursion is when the algorithm uses linear recursion and the algorithm only makes a recursive call as its last operation. For example, ReverseArray() uses tail recursion and LinearSum() does not

6. 6 Linear Recursion Reverse an Array by Iterative non-Recursion Algorithm IterativeReverseArray(A, i, n): Input: An integer array A and int i, n Output: The reversal of n integers in A starting at index i while n > 1 do Swap A[i] and A[i+n-1] i  i + 1 n  n - 2 return

7. 7 Higher order recursion When an instance of recursion algorithm makes more than a single recursive call Algorithm BinarySum(A, i, n): Input: An int array A and int i and n Output: The sum of the n integers in A starting at I if n = 1 then return A[i] return BinarySum(A, i, [n/2]) + BinarySum(A, i, [n/2], [n/2])

8. 8 Higher order recursion When an instance of recursion algorithm makes more than a single recursive call Algorithm BinaryFib(k) Input: An int k Output: The kth Fibonacci number if k <= 1 then return k else return BinaryFib(k-1) + BinaryFib(k-2)