How to effectively visualize a recursive function?

Question

I'm currently in the process of teaching recursion in a programming class. I noticed how hard it is for my students to grasp the concept of recursion. Is there a nice way to visualize what the function does for the pedagogical purposes?

As an example, here is an R function for getting the n'th Fibonacci number:

fib_r <- function(n) {
    if(n <= 2) return(1)
    fib_r(n-1) + fib_r(n-2)
}

Thanks.

Answer 1

This is how I would go about explaining recursive functions in R:

First, I agree with @AEBilgrau that the factorial is a good example for recursion. (Better than Fibonacci in my opionion.)

Then I would quickly go through the theoretical basis why the factorial can be defined as a recursive function, something simple like

4! = 4 * 3 * 2 * 1 = 4 * 3!

Then you could present them the respective recursive R function

fact = function(x) if (x == 0) return(1) else return(x * fact(x - 1))
fact(3)
#6

but present them also the following output

#|fact(3) called
#|fact(3) is calculated via 3*fact(2)
#|fact(2) is unknown yet. Therefore calling fact(2) now
#|Waiting for result from fact(2)
#|  fact(2) called
#|  fact(2) is calculated via 2*fact(1)
#|  fact(1) is unknown yet. Therefore calling fact(1) now
#|  Waiting for result from fact(1)
#|  |   fact(1) called
#|  |   fact(1) is calculated via 1*fact(0)
#|  |   fact(0) is unknown yet. Therefore calling fact(0) now
#|  |   Waiting for result from fact(0)
#|  |   |   fact(0) called
#|  |   |   fact(0)=1 per definition. Nothing to calculate.
#|  |   |   fact(0) returning 1 to waiting fact(1)
#|  |   fact(1) received 1 from fact(0)
#|  |   fact(1) can now calculate 1*fact(0)=1*1=1
#|  |   fact(1) returning 1 to waiting fact(2)
#|  fact(2) received 1 from fact(1)
#|  fact(2) can now calculate 2*fact(1)=2*1=2
#|fact(3) received 2 from fact(2)
#|fact(3) can now calculate 3*fact(2)=3*2=6
#[1] 6

as derived from

# helper function for formatting
tabs = function(n) paste0("|", rep("\t", n), collapse="")

fact = function(x) {
  # determine length of call stack
  sfl = length(sys.frames()) - 1
  # we need to define tmp and tmp1 here because they are used in on.exit
  tmp = NULL
  tmp1 = NULL
  
  # on.exit will print the returned function value when we exit the function ...
  # ... i.e., when one function call is removed from the stack
  on.exit({
    if (sfl > 1) {
      cat(tabs(sfl), "fact(", x, ") returning ",
          tmp, " to waiting fact(", x + 1, ")\n", sep="")
    }
  })
  cat(tabs(sfl), "fact(", x, ") called\n", sep="")
  if (x == 0) {
    cat(tabs(sfl), "fact(0)=1 per definition. Nothing to calculate.\n", sep="")
    # set tmp for printing in on.exit
    tmp = 1
    return(1)
  } else {
    # print some info for students
    cat(tabs(sfl), "fact(", x,
        ") is calculated via ", x, "*fact(", x - 1, ")\n", sep="")
    cat(tabs(sfl),"fact(",x - 1,
        ") is unknown yet. Therefore calling fact(",
        x - 1, ") now\n", sep="")
    cat(tabs(sfl), "Waiting for result from fact(",
        x - 1, ")\n", sep="")
    #call fact again
    tmp1 = fact(x - 1)
    #more info for students
    cat(tabs(sfl), "fact(", x, ") received ", tmp1,
        " from fact(", x - 1, ")\n", sep="")
    tmp = x * tmp1
    cat(tabs(sfl), "fact(", x, ") can now calculate ",
        x, "*fact(", x - 1, ")=", x, "*", tmp1,
        "=", tmp, "\n", sep="")
    return(tmp)
  }
}

fact(3)

Answer 2

Here's my example, probably used in quite a few textbooks:

recursive_sum <- function(n){
  if(n == 1) {print("Remember 1, add everything together"); return(n)}
  print(paste0("Remember ", n, ", pass ", n-1, " to recursive function"))
  n + recursive_sum(n-1)
}

Output:

> recursive_sum(4)
[1] "Remember 4, pass 3 to recursive function"
[1] "Remember 3, pass 2 to recursive function"
[1] "Remember 2, pass 1 to recursive function"
[1] "Remember 1, add everything together"
[1] 10

Answer 3

I think the factorial function is a good example for recursion. Combining this with a printout (as others suggest) seem like a good way to describe what is going on:

factorial <- function(n) {
  cat("factorial(", n, ") was called.\n", sep = "")
  if (n == 0) {
    return(1)
  } else {
    return(n * factorial(n - 1))
  }
}

factorial(4)
#factorial(4) was called.
#factorial(3) was called.
#factorial(2) was called.
#factorial(1) was called.
#factorial(0) was called.
#[1] 24

You can also then implement a non-recursive factorial function and compare the computational efficiencies. Or maybe ask them what is problematic with the above implementation (e.g what happens with factorial(-4)).

Regarding a more proper visualization (and not just easy examples), there are websites which illustrate the recursion tree.

Edit: Googling recursion is also a useful lesson.