尾递归函数，用于在Ocaml中查找树的深度

Question

I have a type tree defined as follows

type 'a tree = Leaf of 'a | Node of 'a * 'a tree * 'a tree ;;

I have a function to find the depth of the tree as follows

let rec depth = function 
    | Leaf x -> 0
    | Node(_,left,right) -> 1 + (max (depth left) (depth right))
;;

This function is not tail recursive. Is there a way for me to write this function in tail recursive way?

Answer 1

You can trivially do this by turning the function into CPS (Continuation Passing Style). The idea is that instead of calling depth left , and then computing things based on this result, you call depth left (fun dleft -> ...) , where the second argument is "what to compute once the result ( dleft ) is available".

let depth tree =
  let rec depth tree k = match tree with
    | Leaf x -> k 0
    | Node(_,left,right) ->
      depth left (fun dleft ->
        depth right (fun dright ->
          k (1 + (max dleft dright))))
  in depth tree (fun d -> d)

This is a well-known trick that can make any function tail-recursive. Voilà, it's tail-rec.

The next well-known trick in the bag is to "defunctionalize" the CPS result. The representation of continuations (the (fun dleft -> ...) parts) as functions is neat, but you may want to see what it looks like as data. So we replace each of these closures by a concrete constructor of a datatype, that captures the free variables used in it.

Here we have three continuation closures: (fun dleft -> depth right (fun dright -> k ...)) , which only reuses the environment variables right and k , (fun dright -> ...) , which reuses k and the now-available left result dleft , and (fun d -> d) , the initial computation, that doesn't capture anything.

type ('a, 'b) cont =
  | Kleft of 'a tree * ('a, 'b) cont (* right and k *)
  | Kright of 'b * ('a, 'b) cont     (* dleft and k *)
  | Kid

The defunctorized function looks like this:

let depth tree =
  let rec depth tree k = match tree with
    | Leaf x -> eval k 0
    | Node(_,left,right) ->
      depth left (Kleft(right, k))
  and eval k d = match k with
    | Kleft(right, k) ->
      depth right (Kright(d, k))
    | Kright(dleft, k) ->
      eval k (1 + max d dleft)
    | Kid -> d
  in depth tree Kid
;;

Instead of building a function k and applying it on the leaves ( k 0 ), I build a data of type ('a, int) cont , which needs to be later eval uated to compute a result. eval , when it gets passed a Kleft , does what the closure (fun dleft -> ...) was doing, that is it recursively call depth on the right subtree. eval and depth are mutually recursive.

Now look hard at ('a, 'b) cont , what is this datatype? It's a list!

type ('a, 'b) next_item =
  | Kleft of 'a tree
  | Kright of 'b

type ('a, 'b) cont = ('a, 'b) next_item list

let depth tree =
  let rec depth tree k = match tree with
    | Leaf x -> eval k 0
    | Node(_,left,right) ->
      depth left (Kleft(right) :: k)
  and eval k d = match k with
    | Kleft(right) :: k ->
      depth right (Kright(d) :: k)
    | Kright(dleft) :: k ->
      eval k (1 + max d dleft)
    | [] -> d
  in depth tree []
;;

And a list is a stack. What we have here is actually a reification (transformation into data) of the call stack of the previous recursive function, with two different cases corresponding to the two different kinds of non-tailrec calls.

Note that the defunctionalization is only there for fun. In pratice the CPS version is short, easy to derive by hand, rather easy to read, and I would recommend using it. Closures must be allocated in memory, but so are elements of ('a, 'b) cont -- albeit those might be represented more compactly`. I would stick to the CPS version unless there are very good reasons to do something more complicated.

Answer 2

In this case (depth computation), you can accumulate over pairs ( subtree depth * subtree content ) to obtain the following tail-recursive function:

let depth tree =
  let rec aux depth = function
    | [] -> depth
    | (d, Leaf _) :: t -> aux (max d depth) t
    | (d, Node (_,left,right)) :: t ->
      let accu = (d+1, left) :: (d+1, right) :: t in
      aux depth accu in
aux 0 [(0, tree)]

For more general cases, you will indeed need to use the CPS transformation described by Gabriel.

Answer 3

There's a neat and generic solution using fold_tree and CPS - continuous passing style:

let fold_tree tree f acc =
  let loop t cont =
    match tree with
    | Leaf -> cont acc
    | Node (x, left, right) ->
      loop left (fun lacc ->
        loop right (fun racc ->
          cont @@ f x lacc racc))
  in loop tree (fun x -> x)

let depth tree = fold_tree tree (fun x dl dr -> 1 + (max dl dr)) 0