Implementation of Binary Search Tree in Elm

Tree.elm

{-----------------------------------------------------------------

A "Tree" represents a binary tree. A "Node" in a binary tree
always has two children. A tree can also be "Empty". Below I have
defined "Tree" and a number of useful functions.

This example also includes some challenge problems :)

-----------------------------------------------------------------}

import Graphics.Element exposing (..)
import Text


type Tree a
    = Empty
    | Node a (Tree a) (Tree a)


empty : Tree a
empty =
    Empty


singleton : a -> Tree a
singleton v =
    Node v Empty Empty


insert : comparable -> Tree comparable -> Tree comparable
insert x tree =
    case tree of
      Empty ->
          singleton x

      Node y left right ->
          if x > y then
              Node y left (insert x right)

          else if x < y then
              Node y (insert x left) right

          else
              tree


fromList : List comparable -> Tree comparable
fromList xs =
    List.foldl insert empty xs


depth : Tree a -> Int
depth tree =
    case tree of
      Empty -> 0
      Node v left right ->
          1 + max (depth left) (depth right)


map : (a -> b) -> Tree a -> Tree b
map f tree =
    case tree of
      Empty -> Empty
      Node v left right ->
          Node (f v) (map f left) (map f right)


t1 = fromList [1,2,3]
t2 = fromList [2,7,5]


main : Element
main =
    flow down
        [ display "depth" depth t1
        , display "depth" depth t2
        , display "map ((+)1)" (map ((+)1)) t2
        , display "sum" sum t2
        , display "flatten" flatten t2
        , display "isElement" (isElement 7) t2
        , display "sum with fold" (fold (+) 0) t2
        , display "flatten with fold" (fold (::) []) t2
        --, display "isElement with fold" (fold ((==) >> (||)) False) t2
        , display "map with fold" (map' ((+)1)) t2
        , display "depth with fold" depth' t2
        ]

map' : (a -> comparable) -> Tree a -> Tree comparable
map' f tree = fold (\x xs -> insert (f x) xs) empty tree

depth' : Tree a -> Int
depth' = fold (\x n -> 1 + n) 0

display : String -> (Tree a -> b) -> Tree a -> Element
display name f value =
    name ++ " (" ++ toString value ++ ") &rArr;\n    " ++ toString (f value) ++ "\n "
        |> Text.fromString
        |> Text.monospace
        |> leftAligned



sum : Tree number -> number
sum tree =
    case tree of
      Empty -> 0
      Node x left right ->
        x + sum left + sum right

flatten : Tree a -> List a
flatten tree =
  case tree of
    Empty -> []
    Node x left right ->
     flatten left ++ [x] ++ flatten right

isElement : comparable -> Tree comparable -> Bool
isElement a tree =
  case tree of
    Empty -> False
    Node x left right -> 
      if x == a then
        True
      else if a > x then
        isElement a right
      else
        isElement a left
        

fold : (a -> b -> b) -> b -> Tree a -> b
fold fn base tree =
  let
    xs = flatten tree
    fold' f v ys =
      case ys of
      [] -> base
      (y::ys) -> f y (fold' f v ys)
  in
    fold' fn base xs
    

{-----------------------------------------------------------------


Exercises:

(1) Sum all of the elements of a tree.

       sum : Tree number -> number

(2) Flatten a tree into a list.

       flatten : Tree a -> List a

(3) Check to see if an element is in a given tree.

       isElement : a -> Tree a -> Bool

(4) Write a general fold function that acts on trees. The fold
    function does not need to guarantee a particular order of
    traversal.

       fold : (a -> b -> b) -> b -> Tree a -> b

(5) Use "fold" to do exercises 1-3 in one line each. The best
    readable versions I have come up have the following length
    in characters including spaces and function name:
      sum: 16
      flatten: 21
      isElement: 46
    See if you can match or beat me! Don't forget about currying
    and partial application!

(6) Can "fold" be used to implement "map" or "depth"?

(7) Try experimenting with different ways to traverse a
    tree: pre-order, in-order, post-order, depth-first, etc.
    More info at: http://en.wikipedia.org/wiki/Tree_traversal

-----------------------------------------------------------------}

@avinashsmiles, the following comes without testing, but consider:
your code: fold f (fold f (f b n) left) right
should be: fold f (fold f (f n b) left) right