[Haskell-cafe] Red-black trees as a nested datatype

[Jim Apple]

-- Inspired by Chris Okasaki's reference to Ross Patterson's AVL
-- trees as a nested datatype, here are (I think) red-black trees
-- as a nested datatype.
-- ref: http://www.haskell.org/pipermail/haskell/2003-April/011693.html

module RedBlackTree where

{-
  Red-black trees satisfy the following conditions,
  according to Wikipedia:

   1. A node is either red or black.
   2. The root is black.
   3. All leaves are black.
   4. Both children of every red node are black.
   5. Every simple path from a node to a descendant leaf contains the
      same number of black nodes.
-}

data Node a n = Node n a n


{- a is the carrier type: the type of the values contained in the
   nodes.
   r0 and b0 are red and black trees with one more level of black
   nodes than r1 and b1.
-}
data Tree a r0 b0 r1 b1 =
    Zero b1 -- The top node of a tree is black
    -- We recurse by adding one to the number of levels of black nodes
  | Succ (Tree a
                 {- Red trees have black children and reduce the count
                    of black nodes to the descendent leaves by 0 -}
                 (Node a b0)
                 {- Black trees have children of either color and reduce
                    the count of black nodes to the descendent leaves by 1 -}
                 (Node a (Either r1 b1))
                 r0
                 b0)

-- The type for black-rooted trees with two levels of black nodes.
type Black2 a  = Node a (Maybe a)

type RedBlackTree a =
    Tree a
    -- A red tree with two levels of black nodes is just a red node on
    -- top of two black nodes.
    (Node (Black2 a) a) (Black2 a) a ()

-- Jim Apple