new: Solved problems 55 to 60

Problems 54A-60/problem_55.hs

@@ -0,0 +1,13 @@
+data Tree a = Empty | Branch a (Tree a) (Tree a)
+  deriving (Show, Eq)
+
+cbalTree :: Int -> [Tree Char]
+cbalTree 0 = [Empty]
+cbalTree n =
+  if even (n - 1)
+    then [Branch 'x' l r | l <- tree_even, r <- tree_even]
+    else [Branch 'x' l r | l <- tree_odd1, r <- tree_odd2] ++ [Branch 'x' l r | l <- tree_odd2, r <- tree_odd1]
+  where
+    tree_even = cbalTree $ (n - 1) `div` 2
+    tree_odd1 = cbalTree $ (n - 1) `div` 2
+    tree_odd2 = cbalTree $ (n - 1) `div` 2 + 1

Problems 54A-60/problem_56.hs

@@ -0,0 +1,11 @@
+data Tree a = Empty | Branch a (Tree a) (Tree a)
+  deriving (Show, Eq)
+
+isMirror :: Tree a -> Tree a -> Bool
+isMirror Empty Empty = True
+isMirror (Branch _ l1 r1) (Branch _ l2 r2) = isMirror l1 r2 && isMirror r1 l2
+isMirror _ _ = False
+
+symmetric :: (Eq a) => Tree a -> Bool
+symmetric Empty = True
+symmetric (Branch _ l r) = isMirror l r

Problems 54A-60/problem_57.hs

@@ -0,0 +1,21 @@
+data Tree a = Empty | Branch a (Tree a) (Tree a)
+  deriving (Show, Eq)
+
+isMirror :: Tree a -> Tree a -> Bool
+isMirror Empty Empty = True
+isMirror (Branch _ l1 r1) (Branch _ l2 r2) = isMirror l1 r2 && isMirror r1 l2
+isMirror _ _ = False
+
+symmetric :: (Eq a) => Tree a -> Bool
+symmetric Empty = True
+symmetric (Branch _ l r) = isMirror l r
+
+add :: (Ord a) => a -> Tree a -> Tree a
+add x Empty = Branch x Empty Empty
+add x t@(Branch v l r) = case compare x v of
+  LT -> Branch v (add x l) r
+  GT -> Branch v l (add x r)
+  EQ -> t
+
+construct :: (Ord a) => [a] -> Tree a
+construct = foldl (flip add) Empty

Problems 54A-60/problem_58.hs

@@ -0,0 +1,25 @@
+data Tree a = Empty | Branch a (Tree a) (Tree a)
+  deriving (Show, Eq)
+
+isMirror :: Tree a -> Tree a -> Bool
+isMirror Empty Empty = True
+isMirror (Branch _ l1 r1) (Branch _ l2 r2) = isMirror l1 r2 && isMirror r1 l2
+isMirror _ _ = False
+
+symmetric :: (Eq a) => Tree a -> Bool
+symmetric Empty = True
+symmetric (Branch _ l r) = isMirror l r
+
+cbalTree :: Int -> [Tree Char]
+cbalTree 0 = [Empty]
+cbalTree n =
+  if even (n - 1)
+    then [Branch 'x' l r | l <- tree_even, r <- tree_even]
+    else [Branch 'x' l r | l <- tree_odd1, r <- tree_odd2] ++ [Branch 'x' l r | l <- tree_odd2, r <- tree_odd1]
+  where
+    tree_even = cbalTree $ (n - 1) `div` 2
+    tree_odd1 = cbalTree $ (n - 1) `div` 2
+    tree_odd2 = cbalTree $ (n - 1) `div` 2 + 1
+
+symCbalTrees :: Int -> [Tree Char]
+symCbalTrees = filter symmetric . cbalTree

Problems 54A-60/problem_59.hs

@@ -0,0 +1,13 @@
+data Tree a = Empty | Branch a (Tree a) (Tree a)
+  deriving (Show, Eq)
+
+hbalTree :: a -> Int -> [Tree a]
+hbalTree _ 0 = [Empty]
+hbalTree x 1 = [Branch x Empty Empty]
+hbalTree x n =
+  [Branch x l r | l <- minustwocase, r <- minusonecase]
+    ++ [Branch x l r | l <- minusonecase, r <- minusonecase]
+    ++ [Branch x l r | l <- minusonecase, r <- minustwocase]
+  where
+    minusonecase = hbalTree x (n - 1)
+    minustwocase = hbalTree x (n - 2)

Problems 54A-60/problem_60.hs

@@ -0,0 +1,36 @@
+data Tree a = Empty | Branch a (Tree a) (Tree a)
+  deriving (Show, Eq)
+
+maxNodes :: Int -> Int
+maxNodes = subtract 1 . (2 ^)
+
+minNodesSeq :: [Int]
+minNodesSeq = 0 : 1 : zipWith ((+) . (1 +)) minNodesSeq (tail minNodesSeq)
+
+minNodes :: Int -> Int
+minNodes = (!!) minNodesSeq
+
+maxHeight :: Int -> Int
+maxHeight n = subtract 1 $ length $ takeWhile (<= n) minNodesSeq
+
+minHeight :: Int -> Int
+minHeight = ceiling . logBase 2 . fromIntegral . (+ 1)
+
+countNodes :: Tree a -> Int
+countNodes Empty = 0
+countNodes (Branch _ l r) = countNodes l + countNodes r + 1
+
+hbalTree :: a -> Int -> [Tree a]
+hbalTree _ 0 = [Empty]
+hbalTree x 1 = [Branch x Empty Empty]
+hbalTree x n =
+  [Branch x l r | l <- minustwocase, r <- minusonecase]
+    ++ [Branch x l r | l <- minusonecase, r <- minusonecase]
+    ++ [Branch x l r | l <- minusonecase, r <- minustwocase]
+  where
+    minusonecase = hbalTree x (n - 1)
+    minustwocase = hbalTree x (n - 2)
+
+hbalTreeNodes :: a -> Int -> [Tree a]
+hbalTreeNodes _ 0 = [Empty]
+hbalTreeNodes x n = concatMap (filter ((n ==) . countNodes) . hbalTree x) [minHeight n .. maxHeight n]