function recursion chapters python course

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tutorial on function recursion in python

recursion is a powerful programming concept where a function calls itself to solve smaller instances of the same problem. this technique is particularly useful for problems that can be broken down into simpler, similar subproblems, such as calculating factorials, fibonacci sequences, and traversing data structures like trees and graphs.

understanding recursion

a recursive function typically follows two main components:
1. **base case**: this is the condition under which the function will stop calling itself. it prevents infinite recursion and eventually leads to a return value.
2. **recursive case**: this is where the function calls itself with modified arguments, progressively working towards the base case.

example: factorial calculation

the factorial of a non-negative integer \( n \) is the product of all positive integers up to \( n \). it can be defined recursively as:
\( \text{factorial}(n) = n \times \text{factorial}(n-1) \) for \( n 0 \)
\( \text{factorial}(0) = 1 \) (base case)

here's how you can implement this in python:

```python
def factorial(n):
base case
if n == 0:
return 1
recursive case
return n * factorial(n - 1)

testing the factorial function
print(factorial(5)) output: 120
```

example: fibonacci sequence

the fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. the sequence starts with 0 and 1. it can be defined recursively as:
\( \text{fibonacci}(n) = \text{fibonacci}(n-1) + \text{fibonacci}(n-2) \) for \( n 1 \)
\( \text{fibonacci}(0) = 0 \)
\( \text{fibonacci}(1) = 1 \)

here's the implementation in python:

```python
def fibonacci(n):
base cases
if n == 0:
return 0
elif n == 1:
return 1
recursive case
return fibonacci(n - 1) + fibonacci(n - 2)

testing the fibonacci function
print(fibonacci(10)) output: 55
```

key points to consider

1. **performance**: recursive ...

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function recursion
python recursion
recursive functions
recursive algorithms
base case
recursive case
stack memory
function calls
depth of recursion
tail recursion
backtracking
memoization
divide and conquer
problem-solving
programming techniques