生成大小为n的所有二叉树

Question

I'd like to write a function which returns all possible binary trees for any given size.

Unfortunately I have no clue, why my solution always returns an empty list, except for size 1.

allTreesN :: Int -> t -> [Tree t]
allTreesN n t 
    | n == 0 = [ Leaf ]
    | otherwise = [(Node x t y) | i <- [0..n-1], x <- (allTreesN i t),y <- (allTreesN i t), size (Node x t y) == n]

Answer 1

You basically generate all trees of size i both for x and y , and then aim to construct a tree of size n . This will only work if i = 2 *n . But now a second problem occurs: we can never generate a tree of size 1 here, since 1 can not be divided by two. Since we can not generate a tree of size 1 , we thus can not generate trees of size 2 , and so on.

We thus need to make sure we generate trees of the correct size. We can do that by generating a tree of size i , and another one of size ni-1 . If we construct a node of that size, we know for sure that the size of a node carrying these subtrees has size n , so we can even omit the check.

So a correct implementation is:

allTreesN :: Int -> a -> [Tree a]
allTreesN 0 _ = [Leaf]
allTreesN n v = [Node l v r | i <- [0..n-1],
                              l <- allTreesN i v,
                              r <- allTreesN  v]

For example:

Prelude> allTreesN 0 'a'
[Leaf]
Prelude> allTreesN 1 'a'
[Node Leaf 'a' Leaf]
Prelude> allTreesN 2 'a'
[Node Leaf 'a' (Node Leaf 'a' Leaf),Node (Node Leaf 'a' Leaf) 'a' Leaf]
Prelude> allTreesN 3 'a'
[Node Leaf 'a' (Node Leaf 'a' (Node Leaf 'a' Leaf)),Node Leaf 'a' (Node (Node Leaf 'a' Leaf) 'a' Leaf),Node (Node Leaf 'a' Leaf) 'a' (Node Leaf 'a' Leaf),Node (Node Leaf 'a' (Node Leaf 'a' Leaf)) 'a' Leaf,Node (Node (Node Leaf 'a' Leaf) 'a' Leaf) 'a' Leaf]