Since "fold" isn't powerful enough to write a tree pretty-printer with indentation, what high-order combinator is?

Question

Given, for example, the following tree data type:

data Tree a = Node [Tree a] | Leaf a deriving Show
type Sexp = Tree String

How do I express a "pretty" function using an high-order combinator, that prints the tree with proper indentation? For example:

sexp = 
    Node [
        Leaf "aaa", 
        Leaf "bbb",
        Node [
            Leaf "ccc",
            Leaf "ddd",
            Node [
                Leaf "eee",
                Leaf "fff"],
            Leaf "ggg",
            Leaf "hhh"],
        Leaf "jjj",
        Leaf "kkk"]
pretty = ????
main = print $ pretty sexp

I want the result of that program to be:

(aaa 
   bbb 
   (ccc 
       ddd 
       (eee 
           fff) 
       ggg 
       hhh) 
   jjj 
   kkk) 

Here is an incomplete solution, using a "fold" as the combinator, that doesn't implement the indentation:

fold f g (Node children) = f (map (fold f g) children)
fold f g (Leaf terminal) = g terminal
pretty = fold (\ x -> "(" ++ (foldr1 ((++) . (++ " ")) x) ++ ")") show
main = putStrLn $ pretty sexp

It is obviously not possible to write the function I want using fold, since it forgets the tree structure. So, what is a proper high-order combinator that is generic enough to allow me to write the function I want, but less powerful than writing a direct recursive function?

Answer 1

fold is strong enough; the trick is that we'll need to instantiate r as a reader monad of the current indentation level.

fold :: ([r] -> r) -> (a -> r) -> (Tree a -> r)
fold node leaf (Node children) = node (map (fold node leaf) children)
fold node leaf (Leaf terminal) = leaf terminal

pretty :: forall a . Show a => Tree a -> String
pretty tree = fold node leaf tree 0 where

  node :: [Int -> String] -> Int -> String
  node children level = 
    let childLines = map ($ level + 1) children
    in unlines ([indent level "Node ["] ++ childLines ++ [indent level "]"])

  leaf :: a -> Int -> String
  leaf a level = indent level (show a)

  indent :: Int -> String -> String -- two space indentation
  indent n s = replicate (2 * n) ' ' ++ s

Take careful note that I pass an extra parameter to the call to fold. This is the initial state of indentation and it works because with this specialization of r, fold returns a function.

Answer 2

It's simply

onLast f xs = init xs ++ [f (last xs)]

pretty :: Sexp -> String
pretty = unlines . fold (node . concat) (:[]) where
    node  []    = [""]
    node (x:xs) = ('(' : x) : map ("  " ++) (onLast (++ ")") xs)