Preorder and inorder sequences of binary trees. We consider binary trees with nodes that are identified by single lower-case letters, as in the example of problem P67.
a) Write predicates preorder/2 and inorder/2 that construct the preorder and inorder sequence of a given binary tree, respectively. The results should be atoms, e.g. 'abdecfg' for the preorder sequence of the example in problem P67.
b) Can you use preorder/2 from problem part a) in the reverse direction; i.e. given a preorder sequence, construct a corresponding tree? If not, make the necessary arrangements.
c) If both the preorder sequence and the inorder sequence of the nodes of a binary tree are given, then the tree is determined unambiguously. Write a predicate pre_in_tree/3 that does the job.
treeToPreorder :: Tree Char -> String treeToPreorder = preorder where preorder Empty = "" preorder (Branch x l r) = x : preorder l ++ preorder r treeToInorder :: Tree Char -> String treeToInorder = inorder where inorder Empty = "" inorder (Branch x l r) = inorder l ++ x : inorder r -- Given a preorder string produce a binary tree such that its preorder string -- is identical to the given one. preToTree :: String -> Tree Char preToTree "" = Empty preToTree (c:cs) = Branch c Empty (preorderToTree cs) -- Given a preorder and an inorder string with unique node chars produce the -- corresponding binary tree. preInTree :: Monad m => String -> String -> m (Tree Char) preInTree   = return Empty preInTree po@(x:xs) io = do (lio,_:rio) <- return $ break (== x) io (lpo,rpo) <- return $ splitAt (length lio) xs l <- preInTree lpo lio r <- preInTree rpo rio return $ Branch x l r preInTree _ _ = fail "woops"