Draw a tree of recursive calls for this algorithm and count the number ofcalls made by the algorithm.d. Is it a good algorithm for solving this problem?

To address your question effectively, let's first clarify that analyzing a recursive algorithm involves understanding its structure and identifying how many times it calls itself. Here’s how to approach your request:

1. Problem Breakdown

You need to construct a tree representation of the recursive calls made by the algorithm. This will depict how many times the function is called for different inputs until a base case is reached. Additionally, we'll evaluate whether this recursion is efficient or suitable for the problem.

2. Relevant Concepts

• Recursive Algorithm: A method of solving a problem where the solution depends on smaller instances of the same problem.
• Binary Tree Representation: Each node represents a function call, and its children represent subsequent function calls made by that call.
• Base Case: The condition under which the recursive calls stop.

3. Analysis and Detail

1. Construct the call tree: If the recursion is akin to computing Fibonacci numbers (for example), where $F(n) = F(n-1) + F(n-2)$:

   • For $F(5)$, the calls would be:
     • $F(5)$
       • $F(4)$
         • $F(3)$
           • $F(2)$
             • $F(1)$
             • $F(0)$
           • $F(2)$
             • $F(1)$
             • $F(0)$
         • $F(3)$ (as shown previously)
       • $F(2)$
         • $F(1)$
         • $F(0)$

2. Counting Calls: Each level of the tree corresponds to a depth in the recursive calls:

   • In this example, we note that many calls to the same values like $F(2)$ and $F(1)$ occur repeatedly.
   • To count total calls efficiently, we can observe that the naive Fibonacci method will result in $2^n$ calls approximately.

4. Verify and Summarize

For our example of the Fibonacci sequence:

• The total number of calls made by $F(n)$ grows exponentially, creating an inefficient algorithm for larger $n$.
• This means the call tree grows drastically, leading to repetitive calculations.

Final Answer

The number of recursive calls in this naive algorithm for Fibonacci is approximately $O(2^n)$, which makes it inefficient for larger values of $n$. Therefore, while recursive algorithms can be elegant, employing memoization or iterative solutions would be better suited for optimizing performance in such cases. For larger problems, this algorithm is not recommended due to its exponential time complexity.