balance :: BTree32 k a -> (k,a) -> BTree32 k a -> BTree32 k a
balance l x r =if not (propbalan (Node (size l + size r +1) l x r)) then
            if not (prop1 (Node (size l + size r +1) l x r)) then
                if size (subarbolizq r) <  2*(size(subarbolder r)) then
                    singleL (Node (size l + size r +1) l x r)
                else
                    doubleL (Node (size l + size r +1) l x r)
            else if not (prop2 (Node (size l + size r +1) l x r)) then
                if size (subarbolizq l) < 2*(size(subarbolder l)) then
                    singleR (Node (size l + size r +1) l x r)
                else
                    doubleR (Node (size l + size r +1) l x r)
            else
                 Node (size l + size r + 1) l x r
        else
            Node (size l + size r +1 ) l x r


singleL :: BTree32 k a -> BTree32 k a
singleL Nil = Nil
singleL (Node n aa (a1,a2) (Node m h (b1,b2) dd)) = Node n (Node (size aa + size h +1 ) aa (a1,a2) h)  (b1,b2) dd


singleR :: BTree32 k a -> BTree32 k a
singleR Nil = Nil
singleR (Node n (Node m bb (b1,b2) h) (a1,a2) aa) = Node n bb (b1,b2) (Node (size aa + size h +1) h (a1,a2) aa)

doubleL :: BTree32 k a -> BTree32 k a
doubleL Nil = Nil
doubleL (Node n aa (a1,a2) (Node m (Node o bb (c1,c2) cc) (b1,b2) dd)) = Node n (Node (size aa + size bb + 1) aa (a1,a2) bb) (c1,c2) (Node (size cc + size dd +1) cc (b1,b2) dd)


doubleR :: BTree32 k a -> BTree32 k a
doubleR Nil = Nil
doubleR (Node n (Node m bb (b1,b2) (Node o cc (c1,c2) dd)) (a1,a2) aa) = Node n (Node (size bb + size cc +1) bb (b1,b2) cc) (c1,c2) (Node (size dd + size aa +1) dd (a1,a2) aa)

prop1 :: BTree32 k a -> Bool
prop1 Nil = True
prop1 (Node n l (x,y) r) = if (size r <= 3*(size l)) then True else False

prop2 :: BTree32 k a -> Bool
prop2 Nil = True
prop2 (Node n l (x,y) r) = if (size l <= 3*(size r)) then True else False

prop3 :: BTree32 k a -> Bool
prop3 Nil = True
prop3 (Node n l (x,y) r ) = if (size l + size r <= 1) then True else False

propbalan :: BTree32 k a -> Bool
propbalan b = if (prop1 b && prop2 b) || prop3 b then True else False

subarbolizq :: BTree32 k a -> BTree32 k a
subarbolizq Nil = Nil
subarbolizq (Node n l x r) = l

subarbolder :: BTree32 k a -> BTree32 k a
subarbolder Nil = Nil
subarbolder (Node n l x r) = r