Here is a set of steps to help you write recursive functions. If you follow each step you are more likely to figure out what to do. Here are the steps in general:

• Write down several examples of how the function works (include both calls and outputs)
• Decide which of the arguments is the one that is recursively decomposed.
• Figure out the base case input for that argument.
• Figure out the recursion by breaking the non-base case input for that argument into multiple pieces.
• Give a name to each of the pieces, there might be 1, 2, or many pieces.
• Work out a recursive example
  1. Pick an example from the several examples above that will be recursive.
  2. Study the imputs and the result
  3. Choose one of the named pieces of the recursively decomposed recursive argument as an argument of a recursive call. What part of the recursively decomposed argument (from the example) does that name refer to? What do the other names refer to?
  4. What is the answer for that recursive call?
  5. How can you combine the answer from the recursive call with the other named peices to get the answer for the whole problem
• Use the steps from the generic program design recipe (where the structure of the body mimics the structure of the input data) to complete the function definition.
  1. It is almost always the case that the structure of breaking up the recursive argument is the structure you need to mimic.

Here is a concrete example for the list based length function.

• Write down several examples of how the function works (include both calls and outputs)
  length [] --> 0
  length [1,2,3] --> 3
  length [3] -> 1
  
• Decide which of the arguments is the one that is recursively decomposed.
  length has only one argument, so there is little choice, that argument is the one that is recursively decomposed.
• Figure out the base case input for that argument.
  The base case is when the list is the empty list: length []
• Figure out the recursion by breaking the non-base case input for that argument into multiple pieces.
  The decomposition is breaking the input list into the first element and all the rest of the elements.
• Give a name to each of the pieces, there might be 1, 2, or many pieces.
  We'll call the first element 'head' and the rest of the elements 'tail'
• Work out a recursive example
  1. Pick an example from the several examples above that will be recursive.
     I choose the second example: length [1,2,3] --> 3
  2. Study the imputs and the result
     The input is [1,2,3], the result is 3.
  3. Choose one of the named pieces of the recursively decomposed recursive argument as an argument of a recursive call. What part of the recursively decomposed argument (from the example) does that name refer to? What do the other names refer to?
     Since length takes a list as input, only the named tail part makes sense as an argument to the recursive call, since it is the only list. head is an element not a list. The recursive call will look like (length tail). tail cooresponds to [2,3] and head cooresponds to 1.
  4. What is the answer for that recursive call?
     (length tail) --> (length [2,3]) --> 2
  5. How can you combine the answer from the recursive call with the other named peices to get the answer for the whole problem?
     The answer to the whole problem is 3, the pieces I have are
     1. 2 from the recursive call (length tail)
     2. tail which is [2,3]
     3. head which is 1
     I can get 3 by adding 1 to 2 (the recurive call).
• Use the steps from the generic program design recipe (where the structure of the body mimics the structure of the input data) to complete the function definition.
  length [] = 0
  length (head : tail) = 1 + length tail
  
  1. It is almost always the case that the structure of breaking up the recursive argument is the structure you need to mimic.
  
  Note that the structure of the input is either the empty list, or a non empty list with first element 'head' and rest of the list 'tail'. Note how this structure is mimiced in the body of the function definition.

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