# Rose tree

A tree with an arbitrary number of sub-trees per node

## Illustration

Such a tree could carry information in all nodes, in which case we speak of a node-labeled rose tree:

```data NLTree a = NLTree a [NLTree a]
```

For instance:

```sampleNLTree =
NLTree 1 [
NLTree 2 [],
NLTree 3 [NLTree 4 []],
NLTree 5 []]
```

Labeling in a rose tree may also be limited to the leaves, in which case we speak of a leaf-labeled rose tree:

```data LLTree a = Leaf a | Fork [LLTree a]
```

For instance:

```sampleLLTree =
Fork [
Leaf 1,
Fork [Leaf 2],
Leaf 3]
```

For what it matters, we can make the type constructors for rose trees functors and foldable types:

```instance Functor NLTree
where
fmap f (NLTree x ts) = NLTree (f x) (fmap (fmap f) ts)

instance Foldable NLTree
where
foldr f z (NLTree x ts) = foldr f z (x : concat (fmap toList ts))

instance Functor LLTree
where
fmap f (Leaf x) = Leaf (f x)
fmap f (Fork ts) = Fork (fmap (fmap f) ts)

instance Foldable LLTree
where
foldr f z (Leaf x) = x `f` z
foldr f z (Fork ts) = foldr f z (concat (fmap toList ts))
```

The fmap definitions basically push fmap into the subtrees while using the list instance of fmap to process lists of subtrees. The foldr definitions basically reduce foldr on trees to 'foldr' on lists by apply toList on subtrees. Here we note that toList can be defined for any foldable type as follows:

```toList :: Foldable t => t a -> [a]
toList = foldMap (\x->[x])
```

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