algorithm - Can I represent a Red-black tree as binary leaf tree?

Question

I've been playing around with RB tree implementation in Haskell but having difficulty changing it a bit so the data is only placed in the leaves, like in a binary leaf tree:

    /+
   /   
 /+    
/       c
a   b

The internal nodes would hold other information e.g. size of tree, in addition to the color of the node (like in a normal RB tree but the data is held in the leaves ony). I am also not needed to sort the data being inserted. I use RB only to get a balanced tree as i insert data but I want to keep the order in which data is inserted.

The original code was (from Okasaki book):

data Color = R | B
data Tree a = E | T Color (Tree a ) a (Tree a)

insert :: Ord a => a -> Tree a -> Tree a
insert x s = makeBlack (ins s)
    where ins E = T R E x E
        ins (T color a y b) 
            | x < y = balance color (ins a) y b
            | x == y = T color a y b
            | x > y = balance color a y (ins b)
        makeBlack (T _ a y b) = T B a y b

I changed it to: (causing Exception:Non-exhaustive patterns in function ins)

data Color = R | B deriving Show
data Tree a = E | Leaf a | T Color Int (Tree a) (Tree a)

insert :: Ord a => a -> Set a -> Set a
insert x s = makeBlack (ins s)
    where 
        ins E = T R 1 (Leaf x) E
        ins (T _ 1 a E) = T R 2 (Leaf x) a
        ins (T color y a b)
            | 0 < y = balance color y (ins a) b
            | 0 == y = T color y a b
            | 0 > y = balance color y a (ins b)
        makeBlack (T _ y a b) = T B y a b

The original balance function is:

balance B (T R (T R a x b) y c) z d = T R (T B a x b) y (T B c z d)
balance B (T R a x (T R b y c)) z d = T R (T B a x b) y (T B c z d)
balance B a x (T R (T R b y c) z d) = T R (T B a x b) y (T B c z d)
balance B a x (T R b y (T R c z d)) = T R (T B a x b) y (T B c z d)
balance color a x b = T color a x b

which i changed a bit as is obvious from my code above.

Thanks in advance for help :)

EDIT: for the kind of representation I'm looking, Chris Okasaki has suggested I use the binary random access list, as described in his book. An alternative would be to simply adapt the code in Data.Set, which is implemented as weight balanced trees.

Answer 1

Assuming you meant insert :: a -> Tree a -> Tree a, then your error probably stems from having no clause for ins (Leaf a).