99 questions/Solutions/55

From HaskellWiki
< 99 questions‎ | Solutions
Revision as of 08:17, 21 November 2010 by Drb226 (talk | contribs) (adding another way: make all combinations and then filter)

Jump to: navigation, search

(**) Construct completely balanced binary trees

In a completely balanced binary tree, the following property holds for every node: The number of nodes in its left subtree and the number of nodes in its right subtree are almost equal, which means their difference is not greater than one.

Write a function cbal-tree to construct completely balanced binary trees for a given number of nodes. The predicate should generate all solutions via backtracking. Put the letter 'x' as information into all nodes of the tree.

cbalTree :: Int -> [Tree Char]
cbalTree 0 = [Empty]
cbalTree n = let (q, r) = (n - 1) `quotRem` 2
    in [Branch 'x' left right | i     <- [q .. q + r],
                                left  <- cbalTree i,
                                right <- cbalTree (n - i - 1)]

This solution uses a list comprehension to enumerate all the trees, in a style that is more natural than standard backtracking.

The base case is a tree of size 0, for which Empty is the only possibility. Trees of size n == 1 or larger consist of a branch, having left and right subtrees with sizes that sum up to n - 1. This is accomplished by getting the quotient and remainder of (n - 1) divided by two; the remainder will be 0 if n is odd, and 1 if n is even. For n == 4, (q, r) = (1, 1).

Inside the list comprehension, i varies from q to q + r. In our n == 4 example, i will vary from 1 to 2. We recursively get all possible left subtrees of size [1..2], and all right subtrees with the remaining elements.

When we recursively call cbalTree 1, q and r will both be 0, thus i will be 0, and the left subtree will simply be Empty. The same goes for the right subtree, since n - i - 1 is 0. This gives back a branch with no children--a "leaf" node:

> cbalTree 1
[Branch 'x' Empty Empty]

The call to cbalTree 2 sets (q, r) = (0, 1), so we'll get back a list of two possible subtrees. One has an empty left branch, the other an empty right branch:

> cbalTree 2
  Branch 'x' Empty (Branch 'x' Empty Empty),
  Branch 'x' (Branch 'x' Empty Empty) Empty

In this way, balances trees of any size can be built recursively from smaller trees.

Another approach is to create a list of all possible tree structures with a given number of nodes, and then filter that list on whether or not the tree is balanced.

data Tree a = Empty | Branch a (Tree a) (Tree a) deriving (Show, Eq)
leaf x = Branch x Empty Empty

main = putStrLn $ concatMap (\t -> show t ++ "\n") balTrees
    where balTrees = filter isBalancedTree (makeTrees 'x' 4)

isBalancedTree :: Tree a -> Bool
isBlanacedTree Empty = True
isBalancedTree (Branch _ l r) = abs (countBranches l - countBranches r)  1
                                && isBalancedTree l && isBalancedTree r
isBalancedTree _ = False

countBranches :: Tree a -> Int
countBranches Empty = 0
countBranches (Branch _ l r) = 1 + countBranches l + countBranches r

-- makes all possible trees filled with the given number of nodes
-- and fill them with the given value
makeTrees :: a -> Int -> [Tree a]
makeTrees _ 0 = []
makeTrees c 1 = [leaf c]
makeTrees c n = lonly ++ ronly ++ landr
    where lonly  = [Branch c t Empty | t <- smallerTree]
          ronly = [Branch c Empty t | t <- smallerTree]
          landr = concat [[Branch c l r | l <- fst lrtrees, r <- snd lrtrees] | lrtrees <- treeMinusTwo]
          smallerTree = makeTrees c (n-1)
          treeMinusTwo = [(makeTrees c num, makeTrees c (n-1-num)) | num <- [0..n-2]]

While not nearly as neat as the previous solution, this solution uses some generic binary tree methods that could be useful in other contexts.