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.php 
<?php
function factorial($n) {
	if ($n == 0) {
		return 1;
	}
	return $n * factorial($n - 1);
}

$result = factorial(5);
echo $result . "\n";

Complexity

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

Implementation notes

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