Difficulty in writing Red Black Tree in F# - algorithm

Question

I am writing a red black tree in F#.
the code which I have written is below. I am facing 2 problems with this code
The rules of balancing the tree state that when the tree has a XYr or rXY type of imbalance I must recolor the 2 parent nodes and IF the grand parent node is not ROOT of the tree then it should be recolored as well.
The difficulty here is that in the recursive approach I only get the next node to work on.. so its hard to know what is the root node.
IN order to solve the above, I added another integer called height to my Node type (type node = Node of int * int * color). That made my pattern matching code in balanceTree function pretty long... but the problem is that when I recolor the grandparent the tree becomes imbalanced because the grand-grand-parent and grand-parent can be red in color which is not allowed.
Can someone recommend a clean way of resolving the issue.
type Color =
| R
| B
type tree =
| Node of int * Color * tree * tree
| Empty
let countNodes tree =
let rec incrCount = function
| Empty -> 0
| Node(_, _, n1, n2) -> 1 + (incrCount n1) + (incrCount n2)
incrCount tree
let isTreeValid tree =
let getTreeBlackNodeHeight tree =
let rec getNodeHeight acc = function
| Empty -> acc + 1
| Node(_, R, n1, _) -> getNodeHeight acc n1
| Node(_, B, n1, _) -> getNodeHeight (acc + 1) n1
getNodeHeight 0 tree
let isRootNodeBlack = function
| Empty -> true
| Node(_, B, _, _) -> true
| Node(_, R, _, _) -> false
let rec areAllBlackHeightsSame height acc = function
| Empty ->
if (acc + 1) = height then true else false
| Node(_, R, n1, n2) -> areAllBlackHeightsSame height acc n1 && areAllBlackHeightsSame height acc n2
| Node(_, B, n1, n2) -> areAllBlackHeightsSame height (acc + 1) n1 && areAllBlackHeightsSame height (acc + 1) n2
let allRedsMustHaveBlackChildren tree =
let getRootNodeColor = function
| Empty -> Color.B
| Node(_, y, _, _) -> y
let rec checkChildColor = function
| Empty -> true
| Node(_, R, n1, n2) -> getRootNodeColor n1 = Color.B && getRootNodeColor n2 = Color.B && checkChildColor n1 && checkChildColor n2
| Node(_, B, n1, n2) -> (checkChildColor n1) && (checkChildColor n2)
checkChildColor tree
(areAllBlackHeightsSame (getTreeBlackNodeHeight tree) 0 tree) && (isRootNodeBlack tree) && (allRedsMustHaveBlackChildren tree)
let insert x tree =
let rec createNode = function
| Empty -> if (countNodes tree) = 0 then Node(x, B, Empty, Empty) else Node(x, R, Empty, Empty)
| Node(i, c, n1, n2) when x > i -> Node(i, c, n1, (createNode n2))
| Node(i, c, n1, n2) when x < i -> Node(i, c, (createNode n1), n2)
| Node(i, _, _, _) when x = i -> failwith "Node already exists"
| _ -> failwith "unknown"
createNode tree
let colorToggle = function
| (i, B) -> (i, R)
| (i, R) -> (i, B)
let balanceTree tree =
let rec balance = function
| Node(gpv, B, Node(p1v, R, Node(c1v, R, a, b), c), Node(p2v, R, d, e)) -> balance (Node(gpv, B, Node(p1v, B, Node(c1v, R, a, b), c), Node(p2v, B, d, e)))
| Node(gpv, B, Node(p1v, R, a, Node(c2v, R, b, c)), Node(p2v, R, d, e)) -> balance (Node(gpv, B, Node(p1v, B, a, Node(c2v, R, b, c)), Node(p2v, B, e, e)))
| Node(gpv, B, Node(p1v, R, a, b), Node(p2v, R, Node(c1v, R, c, d), e)) -> balance (Node(gpv, B, Node(p1v, B, a, b), Node(p2v, B, Node(c1v, R, c, d), e)))
| Node(gpv, B, Node(p1v, R, a, b), Node(p2v, R, c, Node(c2v, R, d, e))) -> balance (Node(gpv, B, Node(p1v, B, a, b), Node(p2v, B, c, Node(c2v, R, d, e))))
| Node(gpv, B, x4, Node(pv, R, x1, Node(cv, R, x2, x3))) -> balance (Node(pv, B, Node(gpv, R, x4, x1), Node(cv, R, x2, x3)))
| Node(gpv, B, x4, Node(pv, R, Node(cv, R, x1, x2), x3)) -> balance (Node(pv, B, Node(gpv, R, x4, Node(cv, B, x1, x2)), x3))
| Node(gpv, B, Node(pv, R, x1, Node(cv, R, x2, x3)), x4) -> balance (Node(pv, B, x1, Node(gpv, R, Node(cv, R, x2, x3), x4)))
| Node(gpv, B, Node(pv, R, Node(cv, R, x1, x2), x3), x4) -> balance (Node(pv, B, (Node(cv, R, x1, x2)), Node(gpv, R, x3, x4)))
| Node(i, x, n1, n2) -> Node(i, x, (balance n1), (balance n2))
| Empty -> Empty
balance tree
[<EntryPoint>]
let main args =
//let t1 = Node((35, B), Node((20, R), Node((10, B), Node((5, R), Empty, Empty), Empty), Node((25, B), Empty, Empty)), Node((85, R), Node((55, B), Node((40, R), Empty, Empty), Node((70, R), Empty, Empty)), Node((100, B), Empty, Empty)))
let t2 = [1 .. 6] |> List.fold (fun acc i-> insert i acc) Empty
printfn "Is Tree Valid : %b" (isTreeValid t2)
let t3 = balanceTree t2
printfn "is Tree Valid : %b" (isTreeValid t3)
0

Answer 1

Standard ML-style implementation in F# looks like this:
type color = R | B
type 'a tree = E | T of color * 'a tree * 'a * 'a tree
let balance = function
| B, T (R, T (R,a,x,b), y, c), z, d
| B, T (R, a, x, T (R,b,y,c)), z, d
| B, a, x, T (R, T (R,b,y,c), z, d)
| 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))
| col, a, x, b -> T (col, a, x, b)
let insert x s =
let rec ins = function
| E -> T (R,E,x,E)
| T (col,a,y,b) as s ->
if x < y then
balance (col, ins a, y, b)
elif x > y then
balance (col, a, y, ins b)
else
s
match ins s with
| T (_,a,y,b) -> T (B,a,y,b)
| t -> t