# Zipper monad/TravelTree

`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.

## Contents

## 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.