Recursion and Dynamic Programming

Factorial (Recursive)

factorial(0) = 1, otherwise factorial(n) = n * factorial(n - 1). The smallest example of recursion with a single base case.

Algorithm

Canonical input n = 5 descends through six call frames and unwinds through five multiplications. Final result 120.

Basic Implementation

basic.rs 
fn factorial(n: i32) -> i32 {
	if n == 0 {
		return 1;
	}
	n * factorial(n - 1)
}

fn main() {
	let result = factorial(5);
	println!("{}", result);
}

Complexity

  • Time: O(n)
  • Space: O(n) call stack

Implementation notes

  • Rust: same recursive shape as the other languages, with fn factorial(n: i32) -> i32 documenting the integer contract.
  • The replay treats the call stack as a vertical list of frames; descent pushes, unwind pops with the computed multiplication shown.
base case `if n == 0 { return 1; }`
recursive call `n * factorial(n - 1)`