# 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.

## 1 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.

## 2 Functions

### 2.1 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.

### 2.2 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.

### 2.3 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 ()```

## 3 Examples

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

### 3.1 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
```

### 3.2 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)
```

#### 3.2.1 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))
```

## 4 Code

The
ZipperTree
library (download):
```module ZipperTree where

import Control.Monad.State
import Control.Arrow (first, second)

import Zipper

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)

-- Utility Functions
--

-- repeat an action until the predicate becomes false
while :: Monad m => m Bool -> m a -> m [a]
while p act = do
b <- p
if b then liftM2 (:) act (while p act) else return []

-- Movement around the tree
--

-- swap branches
swap :: TravelTree a
swap = modify left' >> liftM struct get where
left' (Loc t (R l c)) = Loc { struct = l,
cxt    = L c t }
left' (Loc t (L c r)) = Loc { struct = r,
cxt    = R t c }

-- move down a level, through the left branch
left :: TravelTree a
left = modify left' >> liftM struct get where
left' (Loc (Leaf _    ) _) = error "Down from leaf"
left' (Loc (Branch l r) c) = Loc { struct = l,
cxt    = L c r }

-- move down a level, through the left branch
right :: TravelTree a
right = modify right' >> liftM struct get where
right' (Loc (Leaf _    ) _) = error "Down from leaf"
right' (Loc (Branch l r) c) = Loc { struct = r,
cxt    = R l c }

-- move to a node's parent
up :: TravelTree a
up = modify up' >> liftM struct get where
up' (Loc _     Top) = error "Up from top"
up' (Loc t (L c r)) = Loc { struct = Branch t r, cxt = c }
up' (Loc t (R l c)) = Loc { struct = Branch l t, cxt = c }

-- move to the top node
top :: TravelTree a
top = while (liftM isChild get) up >> liftM struct get

-- get the Loc corresponding to the top of the tree
-- useful for when calling traverse.
-- e.g. (getTop t) `traverse` myPath
getTop :: Tree a -> TreeLoc a
getTop t = (Loc t Top)

-- Node classification
--

-- is the top node
isTop :: TreeLoc a -> Bool
isTop loc = case loc of
(Loc _ Top) -> True
(Loc _ _  ) -> False

-- is not the top node (i.e. the child of some other node)
isChild :: TreeLoc a -> Bool
isChild = not . isTop

-- is a left branch
isLeft :: TreeLoc a -> Bool
isLeft loc = case loc of
(Loc _ Top    ) -> True
(Loc _ (L _ _)) -> True
(Loc _ (R _ _)) -> False

-- is a right branch
isRight :: TreeLoc a -> Bool
isRight loc = isTop loc || (not \$ isLeft loc)```