-- Author: Danila Fedorin, Thursday, January 10
module HW1 where


-- | Integer-labeled binary trees.
data Tree
   = Node Int Tree Tree   -- ^ Internal nodes
   | Leaf Int             -- ^ Leaf nodes
  deriving (Eq,Show)

-- | An example binary tree, which will be used in tests.
t1 :: Tree
t1 = Node 1 (Node 2 (Node 3 (Leaf 4) (Leaf 5))
                    (Leaf 6))
            (Node 7 (Leaf 8) (Leaf 9))

-- | Another example binary tree, used in tests.
t2 :: Tree
t2 = Node 6 (Node 2 (Leaf 1) (Node 4 (Leaf 3) (Leaf 5)))
            (Node 8 (Leaf 7) (Leaf 9))

-- | Right associative fold.
-- The best way here would be to add a foldable instance for Tree.
-- however, since Tree doesn't take a type parameter, this is not possible.
-- We settle for these functions.
treeFoldr :: (Int -> a -> a) -> a -> Tree -> a
treeFoldr f a (Leaf i) = f i a
treeFoldr f a (Node i l r) = treeFoldr f (f i (treeFoldr f a r)) l

-- | foldr for non-empty lists.
treeFoldr1 :: (Int -> Int -> Int) -> Tree -> Int
treeFoldr1 _ (Leaf i) = i
treeFoldr1 f (Node i l r) = treeFoldr f (f i (treeFoldr1 f r)) l

-- | Left associative fold.
treeFoldl :: (Int -> a -> a) -> a -> Tree -> a
treeFoldl f a (Leaf i) = f i a
treeFoldl f a (Node i l r) = treeFoldl f (f i (treeFoldl f a l)) r

-- | foldl for non-empty lists.
treeFoldl1 :: (Int -> Int -> Int) -> Tree -> Int
treeFoldl1 _ (Leaf i) = i
treeFoldl1 f (Node i l r) = treeFoldl f (f i (treeFoldl1 f l)) r

-- | In-order traversal fold.
treeFold :: (Int -> a -> a) -> a -> Tree -> a
treeFold f a (Leaf i) = f i a
treeFold f a (Node i l r) = f i $ treeFold f (treeFold f a r) l

-- | The integer at the left-most node of a binary tree.
--
--   >>> leftmost (Leaf 3)
--   3
--
--   >>> leftmost (Node 5 (Leaf 6) (Leaf 7))
--   6
--
--   >>> leftmost t1
--   4
--
--   >>> leftmost t2
--   1
--
leftmost :: Tree -> Int
leftmost = treeFoldr1 const

-- | The integer at the right-most node of a binary tree.
--
--   >>> rightmost (Leaf 3)
--   3
--
--   >>> rightmost (Node 5 (Leaf 6) (Leaf 7))
--   7
--
--   >>> rightmost t1
--   9
--
--   >>> rightmost t2
--   9
--
rightmost :: Tree -> Int
rightmost = treeFoldl1 const

-- | Get the maximum integer from a binary tree.
--
--   >>> maxInt (Leaf 3)
--   3
--
--   >>> maxInt (Node 5 (Leaf 4) (Leaf 2))
--   5
--
--   >>> maxInt (Node 5 (Leaf 7) (Leaf 2))
--   7
--
--   >>> maxInt t1
--   9
--
--   >>> maxInt t2
--   9
--
maxInt :: Tree -> Int
maxInt = treeFoldr1 max


-- | Get the minimum integer from a binary tree.
--
--   >>> minInt (Leaf 3)
--   3
--
--   >>> minInt (Node 2 (Leaf 5) (Leaf 4))
--   2
--
--   >>> minInt (Node 5 (Leaf 4) (Leaf 7))
--   4
--
--   >>> minInt t1
--   1
--
--   >>> minInt t2
--   1
--
minInt :: Tree -> Int
minInt = treeFoldr1 min


-- | Get the sum of the integers in a binary tree.
--
--   >>> sumInts (Leaf 3)
--   3
--
--   >>> sumInts (Node 2 (Leaf 5) (Leaf 4))
--   11
--
--   >>> sumInts t1
--   45
--
--   >>> sumInts (Node 10 t1 t2)
--   100
--
sumInts :: Tree -> Int
sumInts = treeFoldr1 (+)


-- | The list of integers encountered by a pre-order traversal of the tree.
--
--   >>> preorder (Leaf 3)
--   [3]
--
--   >>> preorder (Node 5 (Leaf 6) (Leaf 7))
--   [5,6,7]
--
--   >>> preorder t1
--   [1,2,3,4,5,6,7,8,9]
--
--   >>> preorder t2
--   [6,2,1,4,3,5,8,7,9]
--
preorder :: Tree -> [ Int ]
preorder = treeFold (:) []

-- | The list of integers encountered by an in-order traversal of the tree.
--
--   >>> inorder (Leaf 3)
--   [3]
--
--   >>> inorder (Node 5 (Leaf 6) (Leaf 7))
--   [6,5,7]
--
--   >>> inorder t1
--   [4,3,5,2,6,1,8,7,9]
--
--   >>> inorder t2
--   [1,2,3,4,5,6,7,8,9]
--
inorder :: Tree -> [ Int ]
inorder = treeFoldr (:) []


-- | Check whether a binary tree is a binary search tree.
--
--   >>> isBST (Leaf 3)
--   True
--
--   >>> isBST (Node 5 (Leaf 6) (Leaf 7))
--   False
--
--   >>> isBST t1
--   False
--
--   >>> isBST t2
--   True
--
isBST :: Tree -> Bool
isBST (Leaf _) = True
isBST (Node i l r) = i >= value l && i <= value r && isBST l && isBST r
    where
        value (Leaf i) = i
        value (Node i _ _) = i


-- | Check whether a number is contained in a binary search tree.
--   (You may assume that the given tree is a binary search tree.)
--
--   >>> inBST 2 (Node 5 (Leaf 2) (Leaf 7))
--   True
--
--   >>> inBST 3 (Node 5 (Leaf 2) (Leaf 7))
--   False
--
--   >>> inBST 4 t2
--   True
--
--   >>> inBST 10 t2
--   False
--
inBST :: Int -> Tree -> Bool
inBST i = treeFoldr (\v acc -> acc || (v == i)) False