How Do I Accurately Determine The Big O Notation Of A Recursive Algorithm That Has Overlapping Subproblems, Such As The Fibonacci Sequence, When The Recursive Calls Are Not Simply A Straightforward Linear Or Exponential Relationship, But Rather Involve A Combination Of Different Operations That Affect The Time Complexity In Distinct Ways?

To determine the big O notation for a recursive algorithm with overlapping subproblems, such as the Fibonacci sequence, follow these steps:

1. Identify the Recurrence Relation: Express the number of operations ${ T(n) }$ in terms of smaller subproblems. For Fibonacci, this is ${ T(n) = T(n-1) + T(n-2) + O(1) }$.

2. Solve the Recurrence Relation: Use methods like the characteristic equation. For Fibonacci, solving ${ r^2 = r + 1 }$ gives roots related to the golden ratio ${ \phi \approx 1.618 }$.

3. Analyze the Growth Rate: The solution to the recurrence shows that ${ T(n) }$ grows exponentially with base ${ \phi }$.

4. Express in Big O Notation: Since ${ \phi^n }$ is exponential, and ${ \phi < 2 }$, the time complexity can be expressed as ${ O(2^n) }$ for simplicity, even though the exact growth is ${ \Theta(\phi^n) }$.

Answer: The time complexity of the recursive Fibonacci algorithm is ${ O(2^n) }$.

\boxed{O(2^n)}