When we talk about recursive algorithms, there’s something really important we need to consider: space complexity.

This term refers to how much memory an algorithm uses. Recursive solutions can use a lot more memory than iterative ones, and here’s why.

Recursive algorithms need something called a call stack. This is like a list that keeps track of all the times a function calls itself. Every time a function calls itself, it adds another layer to this list, which takes up more memory.

Take Fibonacci numbers as an example. The recursive way to find the $n^{th}$ Fibonacci number takes a lot of time, at $O(2^n)$, and it also uses quite a bit of space – $O(n)$. This is because of the deep layers in the recursion tree.

Now, let’s look at an iterative solution. It only needs a constant amount of space, $O(1)$, because it just uses a few variables to keep track of the results. The iterative method works through each number one at a time, so it doesn’t need a lot of extra memory.

Some people believe that recursion makes the code look cleaner and easier to follow, which can be true. But there’s a catch. If the input size gets too big, the stack can run out of space, which causes a crash. You can see this happen in programming languages like C or Java, where deep recursion can lead to a "stack overflow error."

In some cases, tail recursion can help with space issues. If a programming language has something called tail call optimization, it can reuse the space from the current function for the next call. This could lower the extra memory needed to $O(1)$. But not all languages offer this improvement.

In summary, while recursive algorithms can make problems look neat and tidy, they can also use a lot of memory because of the call stack. It’s important to think about the balance between how easy the code is to read and how much performance you need, especially when memory use is a big deal.