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

Basic Implementation

basic.rb 
def factorial(n)
	if n == 0
		return 1
	end
	n * factorial(n - 1)
end

result = factorial(5)
puts result

The pinned run is factorial(5). The diagrams separate the descent, the base case, and the return values so the stack does not feel invisible.

Step 1 - Descend to the base case

Each call waits for one smaller call until f(0) returns 1.

Call tree for factorial(5): f(5) waits on f(4), down to f(0).f(5)waitsf(4)waitsf(3)waitsf(2)waitsf(1)waitsf(0)base = 1

Step 2 - Base value starts the unwind

The first finished frame is f(0) = 1; f(1) can now compute 1 * 1.

Call stack just before unwind begins.top -> bottomknown returnf(0)1f(1)waitingf(2)waitingf(3)waitingf(4)waitingf(5)waiting

Step 3 - Unwind returns 120

Each frame multiplies its n by the completed smaller result.

Return chain for factorial(5).framecalculationreturnsf(0)base1f(1)1 * 11f(2)2 * 12f(3)3 * 26f(4)4 * 624f(5)5 * 24120

Complexity

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

Implementation notes

  • Ruby: same recursive shape as the other languages, with def factorial(n) documenting the integer contract by convention (Ruby is duck-typed; the lesson contract relies on caller hygiene).
  • 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 then return 1 end`
recursive call `n * factorial(n - 1)`