如何有效地可视化递归函数？

Question

我目前正在编程类中教授递归.我注意到我的学生很难掌握递归的概念.有没有一种很好的方法可视化函数对于教学目的的作用？

例如,这是一个用于获取第n个Fibonacci数的R函数:

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

谢谢.

Answer 1

这就是我将如何解释递归函数R：

首先，我同意 @AEBilgrau 的观点，即阶乘是递归的一个很好的例子。（比我认为的斐波那契更好。）

然后我会快速浏览一下为什么阶乘可以定义为递归函数的理论基础，简单的像

4！= 4*3*2*1 = 4*3！

然后你可以向他们展示各自的递归R函数

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

但也向他们展示以下输出

#|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

源自

#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)