What is the space complexity of the recursive implementation used to find the nth fibonacci term? O(1)O(n) O(n2) O(n3)

Break Down the Problem

1. Understand what the Fibonacci sequence is and how recursive implementations work.
2. Analyze the recursive function that calculates the nth Fibonacci number and its underlying call stack.

Relevant Concepts

1. In a recursive implementation, each function call adds a new layer to the call stack.
2. The Fibonacci sequence is defined as: $F(n) = F(n-1) + F(n-2)$ with base cases $F(0) = 0$ and $F(1) = 1$.

Analysis and Detail

1. The recursive Fibonacci function makes two calls for each non-base case:
   • This creates a binary tree of calls, leading to an exponential number of calls in terms of the depth of recursion.
2. Each recursive call requires additional space on the call stack:
   • The maximum depth of recursion is $n$, leading to $O(n)$ space complexity.
   • Thus, the space complexity arises from the stack space used due to recursive calls, rather than the number of computations.

Verify and Summarize

• The recursive implementation retains $O(n)$ space complexity due to the depth of the recursion rather than the overall number of calls or operations.

Final Answer

The space complexity of the recursive implementation used to find the nth Fibonacci term is O(n).