# Recursion

April 27, 2023

In computer science, **recursion** is a programming technique that involves a function calling itself until a certain condition is met. It is a powerful tool for solving problems that require repetitive tasks, such as traversing data structures like trees and graphs, and for implementing complex algorithms like sorting and searching. Recursion is also used in many areas of mathematics, such as fractals, combinatorics, and number theory.

Recursion is based on the concept of a **base case** and a **recursive case**. The base case is the simplest instance of a problem that can be solved directly without recursion. The recursive case is a more complex instance of the same problem that can be reduced to a simpler version of the problem by invoking the same function recursively. The recursion continues until the base case is reached, at which point the function stops calling itself and returns the solution.

A classic example of recursion is the **factorial** function, which computes the product of all positive integers up to a given number. The factorial of n, denoted as n!, can be defined as n times the factorial of (n-1), as long as n is greater than 1. The base case is when n equals 1, in which case the factorial is simply 1. The recursive case is when n is greater than 1, in which case the factorial is n times the factorial of (n-1). Here is the recursive implementation of the factorial function in JavaScript:

```
function factorial(n) {
if (n === 1) {
return 1;
} else {
return n * factorial(n - 1);
}
}
```

The first line of the function defines the base case, which returns 1 when n equals 1. The second line defines the recursive case, which multiplies n by the factorial of (n-1) and returns the result. This implementation of the factorial function is based on the principle of **tail recursion**, which means that the recursive call is the last operation performed by the function before returning. Tail recursion can be optimized by some compilers and interpreters to avoid stack overflow errors that can occur when a large number of recursive calls are made.

Another example of recursion is the **Fibonacci** sequence, which is a series of numbers in which each number is the sum of the two preceding numbers, starting from 0 and 1. The Fibonacci sequence can be defined recursively as follows:

```
fib(0) = 0
fib(1) = 1
fib(n) = fib(n-1) + fib(n-2) for n > 1
```

The base cases are when n equals 0 and 1, in which case the Fibonacci number is simply 0 and 1, respectively. The recursive case is when n is greater than 1, in which case the Fibonacci number is the sum of the Fibonacci number of (n-1) and (n-2). Here is the recursive implementation of the Fibonacci function in Python:

```
def fibonacci(n):
if n == 0:
return 0
elif n == 1:
return 1
else:
return fibonacci(n-1) + fibonacci(n-2)
```

The first two lines of the function define the base cases, which return 0 and 1, respectively, when n equals 0 or 1. The third line defines the recursive case, which returns the sum of the Fibonacci number of (n-1) and (n-2). This implementation of the Fibonacci function is not tail-recursive, which means that it can cause a stack overflow error when called with a large value of n.

Recursion can be used to solve many other problems, such as traversing a binary tree in depth-first order, searching for a path in a maze, generating permutations of a set, and computing the edit distance between two strings. However, recursion also has some drawbacks that must be taken into account when designing recursive algorithms.

One drawback of recursion is that it can be less efficient than iterative solutions for some problems, especially when the recursion depth is large. Each recursive call adds a new stack frame to the call stack, which consumes memory and can cause a stack overflow error if the stack size limit is exceeded. Iterative solutions, on the other hand, use loops or tail-recursive functions that do not add new stack frames and can be optimized by compilers and interpreters.

Another drawback of recursion is that it can be harder to debug and understand than iterative solutions, especially when the recursion depth is large. Each recursive call creates a new instance of the function with its own set of local variables and parameters, which can make it harder to trace the flow of control and data through the program. Recursive solutions also require a good understanding of the base case and the termination condition to avoid infinite loops and other errors.

Despite these drawbacks, recursion is a powerful and elegant technique for solving many problems in computer science and mathematics. It allows us to express complex algorithms in a concise and intuitive way, and to solve problems that would be difficult or impossible to solve with iterative solutions. Recursion is a fundamental concept in programming and a valuable tool in the toolbox of any programmer or computer scientist.