Zipper monad/TravelTree

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TravelTree is a library based on the Zipper monad which is used for traversing binary trees. Read the documentation for the Zipper monad if you haven't already.

Definition[edit]

data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving (Show, Eq)

data Cxt a = Top
           | L (Cxt  a) (Tree a)
           | R (Tree a) (Cxt  a)
             deriving (Show)

type TreeLoc    a = Loc (Cxt a) (Tree a)
type TravelTree a = Travel (TreeLoc a) (Tree a)

We go with the standard definition of a labelless binary tree. Cxt is for storing the context of an element, and TreeLoc for precisely defining the position of an element within a tree, at the same time as defining the tree itself. See Zipper for an explanation of that idiom.

Functions[edit]

Moving around[edit]

There are five main functions for stringing together TravelTree computations:

left,  -- moves down a level, through the left branch
right, -- moves down a level, through the right branch
swap,  -- moves from a left branch to a right branch, or vice versa
up,    -- moves to the node's parent
top    -- moves to the top node
:: TravelTree a

All five return the subtree at the new location.

Mutation[edit]

The three mutation functions defined by the generic Zipper monad (modifyStruct, getStruct and putStruct) are of course available, but there are no TravelTree-specific mutation functions.

Node classification[edit]

There are four functions you can call to find out what kind of node a given location points to:

isTop,   -- is the location the top node?
isChild, -- is the location the child of some other node (i.e. not the top)?
isLeft,  -- is the location a left branch?
isRight  -- is the location a right branch?
:: TreeLoc a -> Bool

Note that these functions are not monadic but instead take a TreeLoc. The TreeLoc pointing to the current node is stored as the state in a TravelTree computation. Thus to call these functions within a do block, use liftM:

do top <- liftM isTop get
   when top $ right >> return ()

Examples[edit]

The following examples use as the example tree:

t = Branch (Branch (Branch (Leaf 1) (Leaf 2))
                   (Leaf 3))
           (Branch (Leaf 4)
                   (Leaf 5))
The example tree

A simple path[edit]

This is a very simple example showing how to use the movement functions:

leftLeftRight :: TravelTree a
leftLeftRight = do left
                   left
                   right

Result of evaluation:

*Tree> (getTop t) `traverse` leftLeftRight
Leaf 2

Tree reverser[edit]

This is a more in-depth example showing getTree and putTree, but is still rather contrived as it's easily done without the zipper (the zipper-less version is shown below).

The algorithm reverses the tree, in the sense that at every branch, the two subtrees are swapped over.

revTree :: Tree a -> Tree a
revTree t = (getTop t) `traverse` revTree' where
    revTree' :: TravelTree a
    revTree' = do t <- getTree
                        case t of
                          Branch _ _ -> do left
                                           l' <- revTree'
                                           swap
                                           r' <- revTree'
                                           up
                                           putTree $ Branch r' l'
                          Leaf x     -> return $ Leaf x

-- without using the zipper:
revTreeZipless :: Tree a -> Tree a
revTreeZipless (Leaf x)       = Leaf x
revTreeZipless (Branch xs ys) = Branch (revTreeZipless ys) (revTreeZipless xs)

Result of evaluation:

*Tree> revTree $ Branch (Leaf 1) (Branch (Branch (Leaf 2) (Leaf 3)) (Leaf 4))
Branch (Branch (Leaf 4) (Branch (Leaf 3) (Leaf 2))) (Leaf 1)

Generalisation[edit]

Einar Karttunen (musasabi) suggested generalising this to a recursive tree mapper:

treeMap :: (a -> Tree a)                -- what to put at leaves
        -> (Tree a -> Tree a -> Tree a) -- what to put at branches
        -> (Tree a -> Tree a)           -- combinator function
treeMap leaf branch = \t -> (getTop t) `traverse` treeMap' where
    treeMap' = do t <- getTree
                  case t of
                    Branch _ _ -> do left
                                     l' <- treeMap'
                                     swap
                                     r' <- treeMap'
                                     up
                                     putTree $ branch l' r'
                    Leaf x     -> return $ leaf x

revTree is then easy:

revTreeZipper :: Tree a -> Tree a
revTreeZipper = treeMap Leaf (flip Branch)

It turns out this is a fairly powerful combinator. As with revTree, it can change the structure of a tree. Here's another example which turns a tree into one where siblings are sorted, i.e. given a Branch l r, if l and r are leaves, then the value of l is less than or equal to that of r. Also, if one of l or r is a Branch and the other a Leaf, then l is the Leaf and r the Branch:

sortSiblings :: Ord a => Tree a -> Tree a
sortSiblings = treeMap Leaf minLeaves where
    minLeaves l@(Branch _ _) r@(Leaf _       ) = Branch r l
    minLeaves l@(Leaf _)     r@(Branch _ _   ) = Branch l r
    minLeaves l@(Branch _ _) r@(Branch _ _   ) = Branch l r
    minLeaves l@(Leaf x)     r@(Leaf y       ) = Branch (Leaf $ min x y)
                                                        (Leaf $ max x y)

Result of evaluation:

*Tree> sortSiblings t
Branch (Branch (Leaf 3) (Branch (Leaf 1) (Leaf 2))) (Branch (Leaf 4) (Leaf 5))

Code[edit]

The code for the TravelTree library is quite length, so you can just download it. Alternatively, you could download the entire zipper library.