Complexity of recursive factorial program

Question

What\'s the complexity of a recursive program to find factorial of a number n? My hunch is that it might be O(n).

Answer 1

The time-complexity of recursive factorial would be:

factorial (n) {    
    if (n = 0) 
        return 1
    else
        return n * factorial(n-1)
}

So,

The time complexity for one recursive call would be:

T(n) = T(n-1) + 3   (3 is for As we have to do three constant operations like 
                 multiplication,subtraction and checking the value of n in each recursive 
                 call)

     = T(n-2) + 6  (Second recursive call)
     = T(n-3) + 9  (Third recursive call)
     .
     .
     .
     .
     = T(n-k) + 3k
     till, k = n

     Then,

     = T(n-n) + 3n
     = T(0) + 3n
     = 1 + 3n

To represent in Big-Oh notation,

T(N) is directly proportional to n,

Therefore, The time complexity of recursive factorial is O(n). As there is no extra space taken during the recursive calls,the space complexity is O(N).