Question
Define recursion
Identify the components
Create pseuocode for a recursion
Identify the components
Create pseuocode for a recursion
Answers
GPT-4o mini
### Definition of Recursion
Recursion is a programming technique where a function calls itself directly or indirectly in order to solve a problem. The problem is broken down into smaller subproblems that resemble the original problem, and the solution is typically built up from the solutions to these smaller problems.
### Components of Recursion
1. **Base Case:** This is a condition under which the recursion terminates. It provides a simple case that can be solved directly without further recursion.
2. **Recursive Case:** This is the part of the function where the function calls itself with modified arguments. The recursive case breaks the problem down into simpler subproblems or smaller instances of the problem.
3. **Recursive Call:** This is the call made by the function to itself, with a reduced or simpler problem to solve.
### Pseudocode for a Recursive Function
Let's create pseudocode for a simple recursive function: calculating the factorial of a non-negative integer \( n \).
```plaintext
FUNCTION Factorial(n)
IF n == 0 THEN
RETURN 1 // Base case: Factorial of 0 is 1
ELSE
RETURN n * Factorial(n - 1) // Recursive case: n! = n * (n-1)!
END FUNCTION
```
### Explanation of Pseudocode
1. **Base Case:** When \( n \) is equal to 0, the function returns 1. This stops the recursion.
2. **Recursive Case:** If \( n \) is greater than 0, the function returns \( n \) multiplied by the result of `Factorial(n - 1)`. This breaks the problem down into a smaller instance by decrementing \( n \).
### Example Usage
If we call `Factorial(5)`, this is what happens:
- Factorial(5) calls Factorial(4)
- Factorial(4) calls Factorial(3)
- Factorial(3) calls Factorial(2)
- Factorial(2) calls Factorial(1)
- Factorial(1) calls Factorial(0)
- Factorial(0) returns 1
Then the returns are calculated as follows, unwinding the recursive calls:
- Factorial(1) returns 1 * 1 = 1
- Factorial(2) returns 2 * 1 = 2
- Factorial(3) returns 3 * 2 = 6
- Factorial(4) returns 4 * 6 = 24
- Factorial(5) returns 5 * 24 = 120
Thus, `Factorial(5)` results in 120.
Recursion is a programming technique where a function calls itself directly or indirectly in order to solve a problem. The problem is broken down into smaller subproblems that resemble the original problem, and the solution is typically built up from the solutions to these smaller problems.
### Components of Recursion
1. **Base Case:** This is a condition under which the recursion terminates. It provides a simple case that can be solved directly without further recursion.
2. **Recursive Case:** This is the part of the function where the function calls itself with modified arguments. The recursive case breaks the problem down into simpler subproblems or smaller instances of the problem.
3. **Recursive Call:** This is the call made by the function to itself, with a reduced or simpler problem to solve.
### Pseudocode for a Recursive Function
Let's create pseudocode for a simple recursive function: calculating the factorial of a non-negative integer \( n \).
```plaintext
FUNCTION Factorial(n)
IF n == 0 THEN
RETURN 1 // Base case: Factorial of 0 is 1
ELSE
RETURN n * Factorial(n - 1) // Recursive case: n! = n * (n-1)!
END FUNCTION
```
### Explanation of Pseudocode
1. **Base Case:** When \( n \) is equal to 0, the function returns 1. This stops the recursion.
2. **Recursive Case:** If \( n \) is greater than 0, the function returns \( n \) multiplied by the result of `Factorial(n - 1)`. This breaks the problem down into a smaller instance by decrementing \( n \).
### Example Usage
If we call `Factorial(5)`, this is what happens:
- Factorial(5) calls Factorial(4)
- Factorial(4) calls Factorial(3)
- Factorial(3) calls Factorial(2)
- Factorial(2) calls Factorial(1)
- Factorial(1) calls Factorial(0)
- Factorial(0) returns 1
Then the returns are calculated as follows, unwinding the recursive calls:
- Factorial(1) returns 1 * 1 = 1
- Factorial(2) returns 2 * 1 = 2
- Factorial(3) returns 3 * 2 = 6
- Factorial(4) returns 4 * 6 = 24
- Factorial(5) returns 5 * 24 = 120
Thus, `Factorial(5)` results in 120.