Tree to ordered list with tail recursion

I am actually sitting over a hour on a problem and don´t find a solution for it. I have this data type: type 'a tree = Empty | Node of 'a * 'a tree * 'a tree

And i have to find a function which converts a given tree in a ordered list. There is also no invariant like that the left child has to be less then the right. I already found a "normal" recursion solution but not a tail recursive solution. I already thought about to build a unordered list and sort it with List.sort, but this uses a merge sort which is not tail recursive. Maybe someone has a good advice.

Thank you!

If you want to traverse the tree in order and return a list, that means our function inorder must have the type 'a tree -> 'a list.

let rec inorder t =
    match t with
      | Empty -> []
      | Node (v, l, r) -> List.append (inorder l) (v :: (inorder r)) (* ! *)

However List.append is in tail position, not inorder. Another problem is we have two calls to inorder. If we put inorder l in tail position, inorder r could not possibly be in tail position - and vice versa.

A neat way to work around this problem is continuation passing style. We take our function above and convert it into a helper function with an extra parameter for our continuation, return

(* convert to helper function, add an extra parameter *)
let rec loop t return =
    match t with
        | Empty -> ...
        | Node (v, l, r) -> ...

The continuation represents "what to do next", so instead of sending values directly out of our function, we must hand them to the continuation instead. That means for the Empty case, we'll return [] - instead of simply []

let rec loop t return =
    match t with
        | Empty -> return []
        | Node (v, l, r) -> ...

For the Node (v, l, r) case, now that we have an extra parameter we can write our own continuation that informs loop what to do next. So to construct our sorted list, we will need to loop l, then loop r (or vice versa), then we can append them. We'll write our program just like this.

let rec loop t return =
    match t with
        | Empty -> return []
        | Node (v, l, r) ->
            loop l ... (* build the left_result *)
            loop r ... (* build the right_result *)
            return (List.append left_result (v :: right_result))

In this next step, we'll fill in the actual lambda syntax for the continuations.

let rec loop t return =
    match t with
        | Empty -> return []
        | Node (v, l, r) ->
            loop l (fun left ->
            loop r (fun right ->
            return (List.append left (v :: right))))

Last, we define inorder which is a call to loop with the default continuation, identity.

let identity x =
    x

let inorder t =
    let rec loop t return =
        match t with
            | Empty -> return []
            | Node (v, l, r) ->
                loop r (fun right ->
                loop l (fun left ->
                return (List.append left (v :: right))))
    in
    loop t identity

As you can see loop r (fun right -> ...) is in tail position for the Node branch. loop l (fun left -> ...) is in tail position of the first continuation. And List.append ... is in tail position of the second continuation. Provided List.append is a tail-recursive procedure, inorder will not grow the stack.

Note using List.append could be a costly choice for big trees. Our function calls it once per Node. Can you think of a way to avoid it? This exercise is left for the reader.