`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

```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

### Moving around

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

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

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

The following examples use as the example tree:

```t = Branch (Branch (Branch (Leaf 1) (Leaf 2))
(Leaf 3))
(Branch (Leaf 4)
(Leaf 5))
```

### A simple path

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

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

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

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