Tree is a comonad

ComonadTree.hs

λ> import Data.Tree
λ> import Control.Comonad
λ> let t = Node (1::Int) [Node 2 [Node 4 [],Node 5 []],Node 3 [Node 6 [],Node 7 []]]
λ> let draw = putStrLn . drawTree . fmap show
λ> draw t
1
|
+- 2
|  |
|  +- 4
|  |
|  `- 5
|
`- 3
   |
   +- 6
   |
   `- 7

λ> levels t
[[1],[2,3],[4,5,6,7]]
λ> draw (duplicate t) -- Comonad duplicate
Node {rootLabel = 1, subForest = [Node {rootLabel = 2, subForest = [Node {rootLabel = 4, subForest = []},Node {rootLabel = 5, subForest = []}]},Node {rootLabel = 3, subForest = [Node {rootLabel = 6, subForest = []},Node {rootLabel = 7, subForest = []}]}]}
|
+- Node {rootLabel = 2, subForest = [Node {rootLabel = 4, subForest = []},Node {rootLabel = 5, subForest = []}]}
|  |
|  +- Node {rootLabel = 4, subForest = []}
|  |
|  `- Node {rootLabel = 5, subForest = []}
|
`- Node {rootLabel = 3, subForest = [Node {rootLabel = 6, subForest = []},Node {rootLabel = 7, subForest = []}]}
   |
   +- Node {rootLabel = 6, subForest = []}
   |
   `- Node {rootLabel = 7, subForest = []}

λ> :t extend -- extend f === fmap f . duplicate
extend :: Comonad w => (w a -> b) -> w a -> w b
λ> draw $ extend levels t
[[1],[2,3],[4,5,6,7]]
|
+- [[2],[4,5]]
|  |
|  +- [[4]]
|  |
|  `- [[5]]
|
`- [[3],[6,7]]
   |
   +- [[6]]
   |
   `- [[7]]

λ> draw $ extend flatten t
[1,2,4,5,3,6,7]
|
+- [2,4,5]
|  |
|  +- [4]
|  |
|  `- [5]
|
`- [3,6,7]
   |
   +- [6]
   |
   `- [7]

λ> draw $ extend (sum.flatten) t
28
|
+- 11
|  |
|  +- 4
|  |
|  `- 5
|
`- 16
   |
   +- 6
   |
   `- 7

λ>