Welcome to our hands-on lesson in which we explore the fascinating concept of recursion on practice. Throughout this lesson, we'll peel back the layers of three foundational problems - Generating the Fibonacci sequence, Finding the sum of all elements in an array, and Calculating factorials. We'll systematically deconstruct each of these problems, expose the core of each solution, and then rebuild them. If the name Fibonacci reminds you of Fibonacci de Piso, one of the most famous mathematicians of the Middle Ages renowned for his work on the Fibonacci sequence, you're not mistaken. Buckle up as we'll retrace this journey of Fibonacci.

So, what is the Fibonacci sequence? The Fibonacci sequence is an infinite sequence of numbers, starting with 0 and 1, with every subsequent number being the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, ...

Given a number n, our task is to implement a recursive algorithm that returns the n-th number in the Fibonacci sequence. Remember, indexing starts from 0, just like Python's 0-index style.

Visualizing Fibonacci might be a little bit tricky at first. If you are into art and design, you might have heard about the golden spiral, a logarithmic spiral whose growth factor is linked to the golden ratio. Interestingly, the Fibonacci sequence is connected with the golden ratio as well. If you look at the ratio of two consecutive numbers in the Fibonacci series, you'll observe that it approximates the golden ratio! It's truly fascinating to see such a foundational mathematical pattern connected to art and design.

Here, we present a straightforward recursive method that returns the n-th Fibonacci number:

Python
Copy to clipboard
1def fib(n): 
2   if n <= 1: 
3       return n 
4   else: 
5       return fib(n - 1) + fib(n - 2)

The above naive recursive solution is a clear and straightforward way, but it is incredibly inefficient. Its inefficiency arises due to repeated calculations.

This inefficiency can be seen by looking at how many times the program calculates fib(n - 2). This function is called once in the scope of fib(n) and then gets called again in the scope of fib(n - 1). This redundancy in calculations greatly bloats up the time required to execute the program, leading to this solution's inefficiency.

For instance, when calculating the 5th Fibonacci number, the 3rd Fibonacci number will be calculated twice, and the 2nd Fibonacci number will be calculated three times. You can even see that, in general, the n-th Fibonacci number will have to calculate the (n - 2)-nd Fibonacci number (n - 1) times, which is incredibly wasteful and inefficient.

In fact, this approach has an exponential time complexity of $O(2^n)$, which means the runtime doubles with each addition to the input size.

Ideally, we would want to reduce this to $O(n)$, which signifies linear time complexity, a much more efficient rate. This can be achieved.

Beyond handling base cases in the naive solution ($n \le 1$), we integrate an optimization technique known as memoization to further enhance the function. With memoization, we store the results of previous Fibonacci number calculations. This way, for any given input n, if we've already calculated the Fibonacci number for n, there's no need to recalculate; the function can directly return the stored result.

This combination of addressing base cases and leveraging memoization avoids unnecessary repetitions and improves the function's efficiency, particularly for larger inputs.

Let's look at how memorization can improve our solution:

Python
Copy to clipboard
1def fib(n, computed={0: 0, 1: 1}):
2    if n not in computed:
3        computed[n] = fib(n - 1, computed) + fib(n - 2, computed)
4    return computed[n]

In this function, we use a Python dictionary computed to store the Fibonacci numbers we've computed so far. If a number n is not in the dictionary computed, we calculate it using the recursive formula and store the result in our dictionary. If the number n is already in our dictionary, we simply return the value, thus avoiding the computation. Edge cases results (n = 0 and n = 1) are stored in the computed list by default.

This function now computes the nth Fibonacci number in linear time $O(n)$, which is a significant improvement over the naive recursive implementation. Each number is calculated only once, and the function can handle a large n efficiently.

Next on our list is a simple operation we frequently perform in our daily lives, which is finding the sum of numbers. Our task is to implement a recursive algorithm that calculates the sum of all elements in an array.

You must be thinking, "Well, adding numbers is easy. I do it all the time when I'm grocery shopping or counting the days left until the next holiday, right?" Absolutely! We regularly sum up numbers. But what about numbers in an array? Imagine trying to add up pages of a manuscript. Instead of summing each page as you go, which can be repetitive and prone to error, you'd sum up the first page with the total pages of the remaining manuscript. Sounds much more manageable, doesn't it?

For-loop Approach: Sure, directly adding up elements using a loop is straightforward. However, just like in the case of the Fibonacci sequence, it misses the significance of understanding and appreciating recursion. So, let's adhere to our recursive path.

Recursive Approach Explanation: In our recursive approach, we decompose our problem similarly. We consider the sum of the array as the sum of the first element and the sum of the remaining elements. It's almost as if we're peeling the array one by one and summing it up, layer by layer.

Instead of considering the base case as an array with a single element, we modify our approach to use an auxiliary index, index. We initially set it to 0, pointing at the first element in the array. The base case is now when index equals the length of the array. At this point, we have processed all elements in the array, and the function should return 0 as there are no more elements to add.

For arrays with more than one element, the function effortlessly adds the current element, pointed at by the index, and the sum of the remaining elements. It achieves this sum by recursively calling the arraySum function, now passing index + 1 as the index. This incremented index moves the 'pointer' to the next element. This way, we can visualize it as moving along the array, adding one element at a time until we reach the end.

This index-based approach eliminates the need for slicing the array, making it more space-efficient. With these modifications, our final solution is:

Python
Copy to clipboard
1def arraySum(arr, index=0): 
2   if index == len(arr): 
3       return 0 
4   else:
5       return arr[index] + arraySum(arr, index + 1)

With this approach, we converge to our sum in a highly efficient and elegant manner without creating unnecessary arrays, ultimately walking our array down to the base case.

Last but not least, we tackle the last task of computing the factorial of a number. The factorial of a number, as many of you would remember from your school math classes, is the product of all positive integers less than or equal to that number, i.e. $n! = n \cdot (n - 1) \cdot (n - 2) \cdot ... \cdot 2 \cdot 1$.

The factorial pops up in the least expected places - in mathematical formulas, treks (when calculating the number of ways to traverse a path), and even poker games (when calculating the number of ways to arrange cards).

Let's revisit this traditional math concept with our new friend, recursion. For example, if we want to compute 5!, one way to do so is $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$. Now, let's flip it. We could also express $5!$ as $5 \cdot 4!$, and $4!$ as $4 \cdot 3!$ and so on until $1!$ or $0!$, which are both equal to 1. There you have it - you've just discovered the recursive nature of the factorial function.

As with the other problems, we first tackle the base cases. If n is 0 or 1, the function unflinchingly returns 1. For n greater than 1, the function humbly returns the product of n and the factorial of (n - 1), by calling itself, i.e., the factorial function, recursively acting on (n - 1). And voila, we have a recursive solution to compute factorials.

Here's the recursive function that calculates the factorial of a number:

Python
Copy to clipboard
1def factorial(n): 
2   if n == 0 or n == 1: 
3       return 1
4   else:
5       return n * factorial(n - 1)

By exploring three fundamental problems that embody the essence of recursion, we realize that recursion is everywhere. Not just in the Fibonacci sequence or factorials, recursion is in the storybook you read, where one story leads to another, and so forth, until you're lost in a maze of colorful adventures. It's also in your family tree, where one generation gives rise to the next, creating a ripple through time. Through this lesson, we've appreciated the elegant simplicity of recursion and hope that you'll begin to recognize its presence everywhere, too.

Shall we practice recursion now?