# 99 questions/Solutions/55

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(a more efficient version) |
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<haskell> | <haskell> | ||

+ | cbalTree :: Int -> [Tree Char] | ||

cbalTree 0 = [Empty] | cbalTree 0 = [Empty] | ||

− | cbalTree n = [Branch 'x' | + | 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)] | ||

</haskell> | </haskell> | ||

− | + | 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 <tt>Empty</tt> is the only possibility. Trees of size <tt>n == 1</tt> or larger consist of a branch, having left and right subtrees with sizes that sum up to <tt>n - 1</tt>. This is accomplished by getting the quotient and remainder of <tt>(n - 1)</tt> divided by two; the remainder will be 0 if <tt>n</tt> is odd, and 1 if <tt>n</tt> is even. For <tt>n == 4</tt>, <tt>(q, r) = (1, 1)</tt>. | ||

+ | |||

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

+ | |||

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

+ | |||

+ | <haskell> | ||

+ | > cbalTree 1 | ||

+ | [Branch 'x' Empty Empty] | ||

+ | </haskell> | ||

+ | |||

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

+ | |||

+ | <haskell> | ||

+ | > cbalTree 2 | ||

+ | [ | ||

+ | Branch 'x' Empty (Branch 'x' Empty Empty), | ||

+ | Branch 'x' (Branch 'x' Empty Empty) Empty | ||

+ | ] | ||

+ | </haskell> | ||

+ | |||

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

+ | |||

+ | A slightly more efficient version of this solution, which never creates the same tree twice: | ||

+ | |||

+ | <haskell> | ||

+ | cbalTree 0 = [Empty] | ||

+ | cbalTree 1 = [leaf 'x'] | ||

+ | cbalTree n = if n `mod` 2 == 1 then | ||

+ | [ Branch 'x' l r | l <- cbalTree ((n - 1) `div` 2), | ||

+ | r <- cbalTree ((n - 1) `div` 2) ] | ||

+ | else | ||

+ | concat [ [Branch 'x' l r, Branch 'x' r l] | l <- cbalTree ((n - 1) `div` 2), | ||

+ | r <- cbalTree (n `div` 2) ] | ||

+ | </haskell> | ||

+ | |||

+ | ---- | ||

+ | 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. | ||

+ | |||

+ | <haskell> | ||

+ | 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]] | ||

+ | </haskell> | ||

+ | |||

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

## Latest revision as of 04:16, 25 August 2012

(**) 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.

A slightly more efficient version of this solution, which never creates the same tree twice:

cbalTree 0 = [Empty] cbalTree 1 = [leaf 'x'] cbalTree n = if n `mod` 2 == 1 then [ Branch 'x' l r | l <- cbalTree ((n - 1) `div` 2), r <- cbalTree ((n - 1) `div` 2) ] else concat [ [Branch 'x' l r, Branch 'x' r l] | l <- cbalTree ((n - 1) `div` 2), r <- cbalTree (n `div` 2) ]

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.