Recursive algorithms can be really cool, but they also come with some special challenges when it comes to how fast they run and how much memory they use. Let’s break down what I learned about these two important topics.

Time Complexity

• What It Means: When a recursive function calls itself, each call goes onto something called the call stack. This means lots of calls can happen at the same time.
• Big O Notation: The time complexity tells us how long a function will take to finish based on how many times it calls itself. For example, a simple recursive function that calculates the Fibonacci sequence can take a lot of time, with a time complexity of $O(2^n)$. That grows really fast! In contrast, a tail-recursive function, or one that splits the problem up, like merge sort, takes less time, with a time complexity of $O(n \log n)$.

Space Complexity

• Memory Use: Each time a recursive function calls itself, it uses some memory. If it gets really deep, it can use a lot of memory. For example, the simple Fibonacci function can use $O(n)$ space because it has $n$ frames on the call stack at its highest point.
• Optimizations: You can sometimes make things better by using different methods, like loop-based (iterative) approaches or improving recursive functions with tricks like memoization. This helps keep track of past results so you don’t have to do the same calculations over and over.

Final Thoughts

Using recursive algorithms can be a beautiful way to solve coding problems, but it's important to understand how they use time and memory. Knowing about these concepts with big O notation can really help you as you learn more in computer science!