Auxiliary Space with Recursive Functions

Prerequisite: Recursion
Memory used by a program is sometimes as important as running time, particularly in constrained environments such as mobile devices. 
For example if we need to create an array of size n, it will require O(n) space. If we need a two-dimensional array of size n x n , it will require O(n²). 
Stack space in recursive calls counts too as extra space required by a program. 
For example : 

C++

int sum(int n)
{
   if (n <= 0)
       return 0;
   return n + sum(n-1);
}

Java

static int sum(int n)
{
   if (n <= 0)
       return 0;
   return n + sum(n - 1);
}

// This code is contributed by Pratham76

Python3

def sum(n):
    if (n <= 0):
        return 0;
    return n + sum(n - 1);


# This code is contributed by Rajput-Ji

C#

static int sum(int n)
{
   if (n <= 0)
       return 0;
   return n + sum(n - 1);
}

// This code is contributed by rutvik_56

Javascript

<script>

function sum(n)
{
   if (n <= 0)
       return 0;
   return n + sum(n - 1);
}

</script>

In the above example function, each call adds a new level to the stack. 

     Sum(5)
    ->sum(4)
            ->sum(3)
            ->sum(2)
                ->sum(1)
                       ->sum(0)

Each of these calls is added to the call stack and takes up actual memory. So code like this would take O(n) time and O(n) auxiliary space. 
However, just because you have n calls total doesn’t mean it takes O(n) space. Consider the below functions, which adds adjacent elements between 0 and n : 
Example: 

CPP

// A non-recursive code that makes n calls
// but takes O(1) extra space.
int pairSumSequence(int n)
{
    int sum = 0;
    for (int i=0; i<n; i++)
        sum += pairSum(i, i+1);
    return sum;
}

int pairSum(int a, int b)
{
    return a + b ;
}

Java

// A non-recursive code that makes n calls
// but takes O(1) extra space.
static int pairSumSequence(int n)
{
    int sum = 0;
    for (int i=0; i<n; i++)
        sum += pairSum(i, i+1);
    return sum;
}

static int pairSum(int a, int b)
{
    return a + b ;
}

// This code is contributed by Pushpesh Raj.

Python3

# A non-recursive code that makes n calls
# but takes O(1) extra space.
def pairSumSequence(n):
    sum = 0
    for i in range(n):
        sum += pairSum(i, i + 1)
    return sum

def pairSum(a, b):
    return a + b

# This code is contributed by Aman Kumar

C#

// A non-recursive code that makes n calls
// but takes O(1) extra space.
static int pairSumSequence(int n)
{
    int sum = 0;
    for (int i = 0; i < n; i++)
        sum += pairSum(i, i+1);
    return sum;
}

static int pairSum(int a, int b)
{
    return a + b ;
}

// This code is contributed by Utkarsh

In this example there will be roughly O(n) calls to pairSum. However, those calls do not exist simultaneously on the call stack, so we need only O(1) space. 
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