Calculating the time and space complexity of a recursive function involves a systematic approach. Here are the steps you can follow to calculate the complexity:

1. Identify the input size: Determine the parameter(s) that affect the size of the input to the recursive function. It could be an integer value, a list, or any other data structure.
2. Define the recursive function: Write down the recursive function and its base case(s). The base case is the condition that stops the recursion and provides a result directly.
3. Define the recurrence relation: Express the time complexity of the function in terms of its recursive calls. This relation represents how the function’s time complexity depends on the input size and the complexity of its recursive calls.
4. Solve the recurrence relation: To solve the recurrence relation, you need to figure out a way to find a simple formula or estimate for the relationship between the inputs and the number of times the function calls itself. This can be done using different methods like substitution, drawing a recursion tree, or using the master theorem. These techniques help you understand how the function behaves and how its complexity grows with the input size.
5. Determine the time complexity: After you have figured out a formula or an approximation for how the function behaves based on the recurrence relation, you can use that information to determine the time complexity of the recursive function. Time complexity tells you how the running time of the function grows as the input size increases. It is usually expressed using Big O notation, which gives an upper bound on the function’s time complexity.
6. Analyze the space complexity: Consider the memory requirements of the recursive function. Identify the variables or data structures that consume memory and analyze how their size grows with the input. Determine the space complexity in terms of the input size, usually using Big O notation.

Let’s walk through an example to calculate the time and space complexity of a recursive function. Consider the following recursive function in Python that calculates the nth Fibonacci number:

def fibonacci(n):
    if n <= 1:
        return n
    else:
        return fibonacci(n-1) + fibonacci(n-2)

Now, let’s go through the steps to calculate its time and space complexity:

Step 1: Identify the input size: In this case, the input size is the value of ‘n’, which represents the index of the Fibonacci number we want to compute.

Step 2: Define the recursive function: We have already defined the Fibonacci function with a base case when n is less than or equal to 1.

Step 3: Define the recurrence relation: The time complexity of the Fibonacci function can be expressed using the recurrence relation T(n) = T(n-1) + T(n-2) + O(1), where T(n) represents the time complexity of computing the nth Fibonacci number.

Step 4: Solve the recurrence relation: We can solve the recurrence relation using different techniques. In this case, we can see that the function makes two recursive calls with smaller inputs (n-1 and n-2). To solve this, we can use the technique of a recursion tree or apply the master theorem.

Applying the master theorem is not straightforward in this case because the Fibonacci function does not follow the required format. Therefore, we’ll use the recursion tree method to analyze the time complexity.

Step 5: Determine the time complexity: By analyzing the recursion tree, we can observe that the function makes two recursive calls at each level, and the depth of the tree is n. Each level of the tree roughly doubles the number of function calls compared to the previous level.

The number of function calls at each level can be represented as a Fibonacci sequence, which grows exponentially. Hence, the time complexity of the Fibonacci function can be approximated as O(2^n) or exponential time complexity.

Step 6: Analyze the space complexity: In terms of space complexity, the function uses memory to store the recursive function calls on the stack. At each recursive call, additional memory is allocated for the function call stack frame.

The maximum depth of the function call stack is also n, as the function makes two recursive calls with smaller inputs. Therefore, the space complexity of the Fibonacci function is O(n), indicating a linear space complexity.

To summarize:

• Time complexity: O(2^n) (exponential)
• Space complexity: O(n) (linear)

Keep in mind that this example showcases a straightforward analysis of the Fibonacci function. More complex recursive functions may require a more intricate analysis, and the time and space complexity may vary accordingly.