How does one identify a problem that requires a recursive solution?

A problem requires a recursive solution when it can be broken down into smaller, similar sub-problems until a base case is reached.

In more detail, recursion is a method of solving problems that involves breaking a problem down into smaller and smaller sub-problems until you get to a small enough problem that it can be solved trivially. Usually, recursion involves a function calling itself while doing so. It is used in situations where the same problem can be divided into smaller instances of the same problem.

To identify a problem that requires a recursive solution, you need to look for problems that can be solved by solving smaller versions of the same problem. For example, problems that involve traversing a tree or a graph data structure often require a recursive solution, as you can solve the problem by solving the same problem on the tree's or graph's sub-structures.

Another indicator is when the problem can be defined in terms of itself. For instance, the Fibonacci sequence is a classic example of a problem that can be solved recursively. Each number in the sequence is the sum of the two preceding ones. So, the problem of calculating a Fibonacci number can be defined in terms of calculating smaller Fibonacci numbers.

Also, problems that require backtracking, such as searching or sorting problems, often require recursion. This is because you can solve these problems by trying out all possible solutions and undoing a step if it doesn't lead to a solution, which is a process that can be defined recursively.

However, it's important to note that just because a problem can be solved recursively doesn't mean it should be. Recursion can lead to solutions that are elegant and easy to understand, but it can also lead to inefficient solutions and stack overflow errors if not used carefully. Therefore, always consider the trade-offs before deciding to use recursion.