Here are the key explanation of Recursive Definitions:

**Recursion **is a process in which the function calls itself indirectly or directly in order to solve the problem. The function that performs the process of recursion is called a **Recursive function**.

• A **recursive data structure** is a data structure that is partially composed of smaller or simpler instances of the same data structure.

• This technique is often used in designing algorithms where a function calls itself to solve smaller instances of the same problem until a base case is reached.

This helps in breaking down complex problems into simpler subproblems.

**The recursive method comprises the following:**

- A base case where the recursion stops and begins to unwind, and
- A recursive step where the method calls itself.

## Need of Recursion:

Recursion is an amazing technique with the help of which we can reduce the length of our code and make it easier to read and write. It has certain advantages over the iteration technique which will be discussed later.

• A task that can be defined with its similar subtask, recursion is one of the best solutions for it. For example; The Factorial of a number.

## Properties of Recursion:

Recursion is a powerful concept in computer science and mathematics.

**Here are some of its key properties:**

**‣ Self-reference: **

Recursion involves solving problems by breaking them down into smaller instances of the same problem. This self-referential aspect is fundamental to recursion.

**‣ Base case:**

Recursion requires a base case, which is the simplest form of the problem that can be solved directly without further recursion. Base cases prevent infinite recursion and provide termination criteria for recursive algorithms.

**‣ Divide and conquer:**

Recursion often follows the “divide and conquer” approach, where a problem is divided into smaller subproblems that are solved recursively. The solutions to these subproblems are then combined to solve the original problem.

**‣ **Function calls:

Recursion involves function calls to the same function within its definition. Each recursive call creates a new instance of the function, which has its own scope and set of parameters.

**‣ **Stack usage:

Recursion utilizes the call stack to manage function calls. Each recursive call adds a new stack frame to the call stack, which stores information such as local variables, parameters, and return addresses.

**‣ Space complexity: **

Recursive algorithms may have higher space complexity compared to their iterative counterparts due to the overhead of maintaining the call stack. However, tail recursion optimization can mitigate this overhead in some cases.

## Steps involved in Recursion:

Here’s a general outline of the steps involved in a recursive algorithm:

**1.) Identify the base case(s): **

Define one or more base cases that represent the simplest form of the problem and can be solved directly without further recursion. Base cases are essential for preventing infinite recursion and providing termination criteria.

**2.) Define the recursive case(s): **

Describe how the problem can be decomposed into smaller, similar subproblems. This step involves identifying the relationship between the original problem and its subproblems and defining the recursive calls accordingly.

**3.) Make recursive calls: **

Within the recursive case(s), make one or more recursive calls to solve the smaller subproblems. Each recursive call should involve a simpler or smaller instance of the original problem, moving closer to the base case(s).

**4.) Combine results (if necessary): **

If the problem requires combining results from recursive calls, define how these results should be combined to obtain the solution to the original problem. This step may involve aggregation, merging, or other operations depending on the problem.

**5.) Handle base case(s): **

When a base case is reached during recursion, provide the solution directly without further recursion. Ensure that the base case(s) cover all possible scenarios and return the appropriate result(s).

**6.) Terminate recursion:**

Ensure that the recursive algorithm terminates properly by ensuring that each recursive call progresses towards a base case. Without proper termination, the algorithm may lead to infinite recursion or stack overflow errors.

**7.) Test and validate:**

Test the recursive algorithm with various inputs to ensure correctness and efficiency. Verify that it produces the expected results for different scenarios and edge cases.

**8.) Optimize (if necessary): **

Analyze the algorithm’s time and space complexity and consider optimization techniques such as memorization, tail recursion optimization, or iterative conversion if needed to improve performance.

## Working of Recursion:

Here’s a simple Java program that demonstrates the working of recursion by calculating the factorial of a given number using a recursive function:

```
public class FactorialCalculator {
// Recursive function to calculate the factorial of a number
public static int factorial(int n) {
// Base case: if n is 0 or 1, factorial is 1
if (n == 0 || n == 1) {
return 1;
} else {
// Recursive case: factorial of n is n multiplied by factorial of (n-1)
return n * factorial(n - 1);
}
}
public static void main(String[] args) {
int number = 5; // Change this number to calculate factorial for a different value
int result = factorial(number);
System.out.println("Factorial of " + number + " is: " + result);
}
}
```

Output:

`Factorial of 5 is: 120`