Step by Step solution about How to resolve the algorithm AVL tree step by step in the Haskell programming language

The provided code defines a data type Tree a to represent a binary search tree, and various functions to manipulate it.

Data Type:

1. data Tree a = Leaf | Node Int (Tree a) a (Tree a):
   • Leaf represents an empty tree.
   • Node represents a tree node with an integer Int for height, a left subtree (Tree a), a value a, and a right subtree (Tree a).

Functions:

1. foldTree :: Ord a => [a] -> Tree a:

   • Folds a list [a] into a binary search tree Tree a using the insert function.

2. height :: Tree a -> Int:

   • Calculates the height of a tree. Leaf has a height of -1, and Node has a height stored in its Int field.

3. depth :: Tree a -> Tree a -> Int:

   • Calculates the depth of the deeper subtree between two trees a and b.

4. insert :: Ord a => a -> Tree a -> Tree a:

   • Inserts a value v into a tree t. If v is less than the current node's value, it goes to the left subtree; if greater, to the right subtree. The tree is then balanced using the rotate function.

5. max_ :: Ord a => Tree a -> Maybe a:

   • Finds the maximum value in a tree. If the tree is empty (Leaf), it returns Nothing; otherwise, it recursively searches the right subtree until it finds a maximum value.

6. delete :: Ord a => a -> Tree a -> Tree a:

   • Deletes a value x from a tree t. If the value is found, the tree is modified accordingly and rebalanced using rotate.

7. rotate :: Tree a -> Tree a:

   • Rotates a tree to maintain balance. It checks for four different cases (Left Left, Right Right, Left Right, Right Left) and applies the appropriate rotation to maintain the height difference between subtrees within 1.

8. draw :: Show a => Tree a -> String:

   • Draw the tree in an ASCII format for visualization. It recursively traverses the tree and constructs a string representation with appropriate padding for each level.

9. main :: IO ():

   • Calls draw on a tree created by foldTree and prints the ASCII representation of the tree.

Example: Suppose we want to create a binary search tree with values [1, 7, 4, 23, 8, 9, 42, 5].

Using the foldTree function:

tree = foldTree [1, 7, 4, 23, 8, 9, 42, 5]

The resulting tree will look like:

  42
 /   \
23   8
/ \   / \
7   9 4   5
\
 1

The ASCII representation generated by draw will be:

       (42,3)
     /       \
   (23,2)    (8,1)
  /   \     /   \
(7,1) (9,0) (4,0) (5,0)
 \
  (1,0)

data Tree a
  = Leaf
  | Node
      Int
      (Tree a)
      a
      (Tree a)
  deriving (Show, Eq)
 
foldTree :: Ord a => [a] -> Tree a
foldTree = foldr insert Leaf
 
height :: Tree a -> Int
height Leaf = -1
height (Node h _ _ _) = h
 
depth :: Tree a -> Tree a -> Int
depth a b = succ (max (height a) (height b))
 
insert :: Ord a => a -> Tree a -> Tree a
insert v Leaf = Node 1 Leaf v Leaf
insert v t@(Node n left v_ right)
  | v_ < v = rotate $ Node n left v_ (insert v right)
  | v_ > v = rotate $ Node n (insert v left) v_ right
  | otherwise = t
 
max_ :: Ord a => Tree a -> Maybe a
max_ Leaf = Nothing
max_ (Node _ _ v right) =
  case right of
    Leaf -> Just v
    _ -> max_ right
 
delete :: Ord a => a -> Tree a -> Tree a
delete _ Leaf = Leaf
delete x (Node h left v right)
  | x == v =
    maybe left (rotate . (Node h left <*> (`delete` right))) (max_ right)
  | x > v = rotate $ Node h left v (delete x right)
  | x < v = rotate $ Node h (delete x left) v right
 
rotate :: Tree a -> Tree a
rotate Leaf = Leaf
rotate (Node h (Node lh ll lv lr) v r)
  -- Left Left.
  | lh - height r > 1 && height ll - height lr > 0 =
    Node lh ll lv (Node (depth r lr) lr v r)
rotate (Node h l v (Node rh rl rv rr))
  -- Right Right.
  | rh - height l > 1 && height rr - height rl > 0 =
    Node rh (Node (depth l rl) l v rl) rv rr
rotate (Node h (Node lh ll lv (Node rh rl rv rr)) v r)
  -- Left Right.
  | lh - height r > 1 =
    Node h (Node (rh + 1) (Node (lh - 1) ll lv rl) rv rr) v r
rotate (Node h l v (Node rh (Node lh ll lv lr) rv rr))
  -- Right Left.
  | rh - height l > 1 =
    Node h l v (Node (lh + 1) ll lv (Node (rh - 1) lr rv rr))
rotate (Node h l v r) =
  -- Re-weighting.
  let (l_, r_) = (rotate l, rotate r)
   in Node (depth l_ r_) l_ v r_
 
draw :: Show a => Tree a -> String
draw t = '\n' : draw_ t 0 <> "\n"
  where
    draw_ Leaf _ = []
    draw_ (Node h l v r) d = draw_ r (d + 1) <> node <> draw_ l (d + 1)
      where
        node = padding d <> show (v, h) <> "\n"
        padding n = replicate (n * 4) ' '
 
main :: IO ()
main = putStr $ draw $ foldTree [1 .. 31]