How does one trace a recursive algorithm step-by-step?

One traces a recursive algorithm step-by-step by following the sequence of function calls and returns, and tracking the changing state of variables.

To trace a recursive algorithm, you need to understand the concept of recursion first. Recursion is a method where the solution to a problem depends on solutions to smaller instances of the same problem. In programming, a function that calls itself is known as a recursive function.

To trace a recursive algorithm, you start by identifying the base case. The base case is the simplest form of the problem that the algorithm can solve directly. It's the condition that stops the recursion from continuing indefinitely.

Next, you need to understand the recursive case. This is the part of the function that calls itself to solve a smaller version of the problem. It's important to note that each recursive call should bring you closer to the base case, otherwise, the recursion would go on indefinitely, leading to a stack overflow error.

Now, to trace the algorithm, you follow the sequence of function calls. Start with the initial call to the function, with the original problem size. Then, follow the sequence of recursive calls that the function makes, each time with a smaller problem size.

For each function call, keep track of the state of all relevant variables. This includes the function's parameters, any local variables, and the return value. You can do this by drawing a recursion tree or stack, where each node or layer represents a function call, and its children or lower layers represent the recursive calls it makes.

When a function call reaches the base case, it will return a value without making any more recursive calls. This return value will then be used by the function call that made the recursive call, and so on, until you get back to the original function call.

By following this process, you can trace the execution of a recursive algorithm step-by-step, and understand how it works to solve the problem. This is a crucial skill for debugging and optimising recursive algorithms, and for understanding more complex algorithms that use recursion.