module Tree where


--
-- * Part 1: Binary trees
--

-- | 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))


-- | 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 (Leaf i)     = i
leftmost (Node _ l _) = leftmost l


-- | 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 (Leaf i) = i
rightmost (Node _ _ r) = rightmost r


-- | 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 (Leaf i) = i
maxInt (Node x l r) = max x (max (maxInt l) (maxInt r))


-- | 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 (Leaf i) = i
minInt (Node x l r) = min x (min (minInt l) (minInt r))


-- | 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 (Leaf i) = i
sumInts (Node x l r) = sum ([x] ++ [sumInts l] ++ [sumInts r])


-- | 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 (Leaf i) = [i]
preorder (Node x l r) = [x] ++ preorder l ++ preorder r

-- | 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 (Leaf i) = [i]
inorder (Node x l r) =  inorder l ++ [x] ++ inorder r


-- | 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 x (Leaf l) (Leaf r))
    | l > x = False
    | r < x = False
    | otherwise = True
isBST (Node x (Node l al ar) (Node r bl br))
    | l > x = False
    | r < x = False
    | isBST al && isBST ar && isBST bl && isBST br == True = True
    | otherwise = True
-- | 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 (Leaf x) = x == i
inBST i (Node x l r) = elem True ([x == i] ++ [inBST i l] ++ [inBST i r])