Data/Tree.hs

    {-# LANGUAGE PatternGuards #-}
    {-# LANGUAGE CPP #-}
    #if __GLASGOW_HASKELL__
    {-# LANGUAGE DeriveDataTypeable, StandaloneDeriving #-}
    {-# LANGUAGE DeriveGeneric #-}
    {-# LANGUAGE Trustworthy #-}
    #endif

    #include "containers.h"

    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Data.Tree
    -- Copyright   :  (c) The University of Glasgow 2002
    -- License     :  BSD-style (see the file libraries/base/LICENSE)
    --
    -- Maintainer  :  libraries@haskell.org
    -- Portability :  portable
    --
    -- = Multi-way Trees and Forests
    --
    -- The @'Tree' a@ type represents a lazy, possibly infinite, multi-way tree
    -- (also known as a /rose tree/).
    --
    -- The @'Forest' a@ type represents a forest of @'Tree' a@s.
    --
    -----------------------------------------------------------------------------

    module Data.Tree(

        -- * Trees and Forests
          Tree(..)
        , Forest

        -- * Construction
        , unfoldTree
        , unfoldForest
        , unfoldTreeM
        , unfoldForestM
        , unfoldTreeM_BF
        , unfoldForestM_BF

        -- * Elimination
        , foldTree
        , flatten
        , levels

        -- * Ascii Drawings
        , drawTree
        , drawForest

        ) where

    #if MIN_VERSION_base(4,8,0)
    import Data.Foldable (toList)
    import Control.Applicative (Applicative(..), liftA2)
    #else
    import Control.Applicative (Applicative(..), liftA2, (<$>))
    import Data.Foldable (Foldable(foldMap), toList)
    import Data.Monoid (Monoid(..))
    import Data.Traversable (Traversable(traverse))
    #endif

    import Control.Monad (liftM)
    import Control.Monad.Fix (MonadFix (..), fix)
    import Data.Sequence (Seq, empty, singleton, (<|), (|>), fromList,
                ViewL(..), ViewR(..), viewl, viewr)
    import Data.Typeable
    import Control.DeepSeq (NFData(rnf))

    #ifdef __GLASGOW_HASKELL__
    import Data.Data (Data)
    import GHC.Generics (Generic, Generic1)
    #endif

    import Control.Monad.Zip (MonadZip (..))

    #if MIN_VERSION_base(4,8,0)
    import Data.Coerce
    #endif

    #if MIN_VERSION_base(4,9,0)
    import Data.Functor.Classes
    import Data.Semigroup (Semigroup (..))
    #endif

    #if !MIN_VERSION_base(4,8,0)
    import Data.Functor ((<$))
    #endif

    -- | Non-empty, possibly infinite, multi-way trees; also known as /rose trees/.
    data Tree a = Node {
            rootLabel :: a,         -- ^ label value
            subForest :: Forest a   -- ^ zero or more child trees
        }
    #ifdef __GLASGOW_HASKELL__
      deriving ( Eq
               , Read
               , Show
               , Data
               , Generic  -- ^ @since 0.5.8
               , Generic1 -- ^ @since 0.5.8
               )
    #else
      deriving (Eq, Read, Show)
    #endif

    type Forest a = [Tree a]

    #if MIN_VERSION_base(4,9,0)
    -- | @since 0.5.9
    instance Eq1 Tree where
      liftEq eq = leq
        where
          leq (Node a fr) (Node a' fr') = eq a a' && liftEq leq fr fr'

    -- | @since 0.5.9
    instance Ord1 Tree where
      liftCompare cmp = lcomp
        where
          lcomp (Node a fr) (Node a' fr') = cmp a a' <> liftCompare lcomp fr fr'

    -- | @since 0.5.9
    instance Show1 Tree where
      liftShowsPrec shw shwl p (Node a fr) = showParen (p > 10) $
            showString "Node {rootLabel = " . shw 0 a . showString ", " .
              showString "subForest = " . liftShowList shw shwl fr .
              showString "}"

    -- | @since 0.5.9
    instance Read1 Tree where
      liftReadsPrec rd rdl p = readParen (p > 10) $
        \s -> do
          ("Node", s1) <- lex s
          ("{", s2) <- lex s1
          ("rootLabel", s3) <- lex s2
          ("=", s4) <- lex s3
          (a, s5) <- rd 0 s4
          (",", s6) <- lex s5
          ("subForest", s7) <- lex s6
          ("=", s8) <- lex s7
          (fr, s9) <- liftReadList rd rdl s8
          ("}", s10) <- lex s9
          pure (Node a fr, s10)
    #endif

    INSTANCE_TYPEABLE1(Tree)

    instance Functor Tree where
        fmap = fmapTree
        x <$ Node _ ts = Node x (map (x <$) ts)

    fmapTree :: (a -> b) -> Tree a -> Tree b
    fmapTree f (Node x ts) = Node (f x) (map (fmapTree f) ts)
    #if MIN_VERSION_base(4,8,0)
    -- Safe coercions were introduced in 4.7.0, but I am not sure if they played
    -- well enough with RULES to do what we want.
    {-# NOINLINE [1] fmapTree #-}
    {-# RULES
    "fmapTree/coerce" fmapTree coerce = coerce
     #-}
    #endif

    instance Applicative Tree where
        pure x = Node x []
        Node f tfs <*> tx@(Node x txs) =
            Node (f x) (map (f <$>) txs ++ map (<*> tx) tfs)
    #if MIN_VERSION_base(4,10,0)
        liftA2 f (Node x txs) ty@(Node y tys) =
            Node (f x y) (map (f x <$>) tys ++ map (\tx -> liftA2 f tx ty) txs)
    #endif
        Node x txs <* ty@(Node _ tys) =
            Node x (map (x <$) tys ++ map (<* ty) txs)
        Node _ txs *> ty@(Node y tys) =
            Node y (tys ++ map (*> ty) txs)

    instance Monad Tree where
        return = pure
        Node x ts >>= f = case f x of
            Node x' ts' -> Node x' (ts' ++ map (>>= f) ts)

    -- | @since 0.5.11
    instance MonadFix Tree where
      mfix = mfixTree

    mfixTree :: (a -> Tree a) -> Tree a
    mfixTree f
      | Node a children <- fix (f . rootLabel)
      = Node a (zipWith (\i _ -> mfixTree ((!! i) . subForest . f))
                        [0..] children)

    instance Traversable Tree where
        traverse f (Node x ts) = liftA2 Node (f x) (traverse (traverse f) ts)

    instance Foldable Tree where
        foldMap f (Node x ts) = f x `mappend` foldMap (foldMap f) ts

    #if MIN_VERSION_base(4,8,0)
        null _ = False
        {-# INLINE null #-}
        toList = flatten
        {-# INLINE toList #-}
    #endif

    instance NFData a => NFData (Tree a) where
        rnf (Node x ts) = rnf x `seq` rnf ts

    instance MonadZip Tree where
      mzipWith f (Node a as) (Node b bs)
        = Node (f a b) (mzipWith (mzipWith f) as bs)

      munzip (Node (a, b) ts) = (Node a as, Node b bs)
        where (as, bs) = munzip (map munzip ts)

    -- | 2-dimensional ASCII drawing of a tree.
    --
    -- ==== __Examples__
    --
    -- > putStr $ drawTree $ fmap show (Node 1 [Node 2 [], Node 3 []])
    --
    -- @
    -- 1
    -- |
    -- +- 2
    -- |
    -- `- 3
    -- @
    --
    drawTree :: Tree String -> String
    drawTree  = unlines . draw

    -- | 2-dimensional ASCII drawing of a forest.
    --
    -- ==== __Examples__
    --
    -- > putStr $ drawForest $ map (fmap show) [(Node 1 [Node 2 [], Node 3 []]), (Node 10 [Node 20 []])]
    --
    -- @
    -- 1
    -- |
    -- +- 2
    -- |
    -- `- 3
    --
    -- 10
    -- |
    -- `- 20
    -- @
    --
    drawForest :: Forest String -> String
    drawForest  = unlines . map drawTree

    draw :: Tree String -> [String]
    draw (Node x ts0) = lines x ++ drawSubTrees ts0
      where
        drawSubTrees [] = []
        drawSubTrees [t] =
            "|" : shift "`- " "   " (draw t)
        drawSubTrees (t:ts) =
            "|" : shift "+- " "|  " (draw t) ++ drawSubTrees ts

        shift first other = zipWith (++) (first : repeat other)

    -- | Returns the elements of a tree in pre-order.
    --
    -- @
    --
    --   a
    --  / \\    => [a,b,c]
    -- b   c
    -- @
    --
    -- ==== __Examples__
    --
    -- > flatten (Node 1 [Node 2 [], Node 3 []]) == [1,2,3]
    flatten :: Tree a -> [a]
    flatten t = squish t []
      where squish (Node x ts) xs = x:Prelude.foldr squish xs ts

    -- | Returns the list of nodes at each level of the tree.
    --
    -- @
    --
    --   a
    --  / \\    => [[a], [b,c]]
    -- b   c
    -- @
    --
    -- ==== __Examples__
    --
    -- > levels (Node 1 [Node 2 [], Node 3 []]) == [[1],[2,3]]
    --
    levels :: Tree a -> [[a]]
    levels t =
        map (map rootLabel) $
            takeWhile (not . null) $
            iterate (concatMap subForest) [t]

    -- | Fold a tree into a "summary" value in depth-first order.
    --
    -- For each node in the tree, apply @f@ to the @rootLabel@ and the result
    -- of applying @f@ to each @subForent@.
    --
    -- This is also known as the catamorphism on trees.
    --
    -- ==== __Examples__
    --
    -- Sum the values in a tree:
    --
    -- > foldTree (\x xs -> sum (x:xs)) (Node 1 [Node 2 [], Node 3 []]) == 6
    --
    -- Find the maximum value in the tree:
    --
    -- > foldTree (\x xs -> maximum (x:xs)) (Node 1 [Node 2 [], Node 3 []]) == 3
    --
    --
    -- @since 0.5.8
    foldTree :: (a -> [b] -> b) -> Tree a -> b
    foldTree f = go where
        go (Node x ts) = f x (map go ts)

    -- | Build a (possibly infinite) tree from a seed value in breadth-first order.
    --
    -- @unfoldTree f b@ constructs a tree by starting with the tree
    -- @Node { rootLabel=b, subForest=[] }@ and repeatedly applying @f@ to each
    -- 'rootLabel' value in the tree's leaves to generate its 'subForest'.
    --
    -- For a monadic version see 'unfoldTreeM_BF'.
    --
    -- ==== __Examples__
    --
    -- Construct the tree of @Integer@s where each node has two children:
    -- @left = 2*x@ and @right = 2*x + 1@, where @x@ is the 'rootLabel' of the node.
    -- Stop when the values exceed 7.
    --
    -- > let buildNode x = if 2*x + 1 > 7 then (x, []) else (x, [2*x, 2*x+1])
    -- > putStr $ drawTree $ fmap show $ unfoldTree buildNode 1
    --
    -- @
    --
    -- 1
    -- |
    -- +- 2
    -- |  |
    -- |  +- 4
    -- |  |
    -- |  `- 5
    -- |
    -- `- 3
    --    |
    --    +- 6
    --    |
    --    `- 7
    -- @
    --
    unfoldTree :: (b -> (a, [b])) -> b -> Tree a
    unfoldTree f b = let (a, bs) = f b in Node a (unfoldForest f bs)

    -- | Build a (possibly infinite) forest from a list of seed values in
    -- breadth-first order.
    --
    -- @unfoldForest f seeds@ invokes 'unfoldTree' on each seed value.
    --
    -- For a monadic version see 'unfoldForestM_BF'.
    --
    unfoldForest :: (b -> (a, [b])) -> [b] -> Forest a
    unfoldForest f = map (unfoldTree f)

    -- | Monadic tree builder, in depth-first order.
    unfoldTreeM :: Monad m => (b -> m (a, [b])) -> b -> m (Tree a)
    unfoldTreeM f b = do
        (a, bs) <- f b
        ts <- unfoldForestM f bs
        return (Node a ts)

    -- | Monadic forest builder, in depth-first order
    unfoldForestM :: Monad m => (b -> m (a, [b])) -> [b] -> m (Forest a)
    unfoldForestM f = Prelude.mapM (unfoldTreeM f)

    -- | Monadic tree builder, in breadth-first order.
    --
    -- See 'unfoldTree' for more info.
    --
    -- Implemented using an algorithm adapted from /Breadth-First Numbering: Lessons
    -- from a Small Exercise in Algorithm Design/, by Chris Okasaki, /ICFP'00/.
    unfoldTreeM_BF :: Monad m => (b -> m (a, [b])) -> b -> m (Tree a)
    unfoldTreeM_BF f b = liftM getElement $ unfoldForestQ f (singleton b)
      where
        getElement xs = case viewl xs of
            x :< _ -> x
            EmptyL -> error "unfoldTreeM_BF"

    -- | Monadic forest builder, in breadth-first order
    --
    -- See 'unfoldForest' for more info.
    --
    -- Implemented using an algorithm adapted from /Breadth-First Numbering: Lessons
    -- from a Small Exercise in Algorithm Design/, by Chris Okasaki, /ICFP'00/.
    unfoldForestM_BF :: Monad m => (b -> m (a, [b])) -> [b] -> m (Forest a)
    unfoldForestM_BF f = liftM toList . unfoldForestQ f . fromList

    -- Takes a sequence (queue) of seeds and produces a sequence (reversed queue) of
    -- trees of the same length.
    unfoldForestQ :: Monad m => (b -> m (a, [b])) -> Seq b -> m (Seq (Tree a))
    unfoldForestQ f aQ = case viewl aQ of
        EmptyL -> return empty
        a :< aQ' -> do
            (b, as) <- f a
            tQ <- unfoldForestQ f (Prelude.foldl (|>) aQ' as)
            let (tQ', ts) = splitOnto [] as tQ
            return (Node b ts <| tQ')
      where
        splitOnto :: [a'] -> [b'] -> Seq a' -> (Seq a', [a'])
        splitOnto as [] q = (q, as)
        splitOnto as (_:bs) q = case viewr q of
            q' :> a -> splitOnto (a:as) bs q'
            EmptyR -> error "unfoldForestQ"