4.5. Beyond Lists

Functionals like map and fold are not restricted to lists. They make sense for nearly any kind of data collection. For example, recall this tree representation:

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

4.5.1. Map on Trees

This one is easy. All we have to do is apply the function f to the value v at each node:

let rec map_tree f = function
  | Leaf -> Leaf
  | Node (v, l, r) -> Node (f v, map_tree f l, map_tree f r)
val map_tree : ('a -> 'b) -> 'a tree -> 'b tree = <fun>

4.5.2. Fold on Trees

This one is only a little harder. Let’s develop a fold functional for 'a tree similar to our fold_right over 'a list. One way to think of List.fold_right would be that the [] value in the list gets replaced by the acc argument, and each :: constructor gets replaced by an application of the f argument. For example, [a; b; c] is syntactic sugar for a :: (b :: (c :: [])). So if we replace [] with 0 and :: with ( + ), we get a + (b + (c + 0)). Along those lines, here’s a way we could rewrite fold_right that will help us think a little more clearly:

type 'a mylist =
  | Nil
  | Cons of 'a * 'a mylist

let rec fold_mylist f acc = function
  | Nil -> acc
  | Cons (h, t) -> f h (fold_mylist f acc t)
type 'a mylist = Nil | Cons of 'a * 'a mylist
val fold_mylist : ('a -> 'b -> 'b) -> 'b -> 'a mylist -> 'b = <fun>

The algorithm is the same. All we’ve done is to change the definition of lists to use constructors written with alphabetic characters instead of punctuation, and to change the argument order of the fold function.

For trees, we’ll want the initial value of acc to replace each Leaf constructor, just like it replaced [] in lists. And we’ll want each Node constructor to be replaced by the operator. But now the operator will need to be ternary instead of binary—that is, it will need to take three arguments instead of two—because a tree node has a value, a left child, and a right child, whereas a list cons had only a head and a tail.

Inspired by those observations, here is the fold function on trees:

let rec fold_tree f acc = function
  | Leaf -> acc
  | Node (v, l, r) -> f v (fold_tree f acc l) (fold_tree f acc r)
val fold_tree : ('a -> 'b -> 'b -> 'b) -> 'b -> 'a tree -> 'b = <fun>

If you compare that function to fold_mylist, you’ll note it very nearly identical. There’s just one more recursive call in the second pattern-matching branch, corresponding to the one more occurrence of 'a tree in the definition of that type.

We can then use fold_tree to implement some of the tree functions we’ve previously seen:

let size t = fold_tree (fun _ l r -> 1 + l + r) 0 t
let depth t = fold_tree (fun _ l r -> 1 + max l r) 0 t
let preorder t = fold_tree (fun x l r -> [x] @ l @ r) [] t
val size : 'a tree -> int = <fun>
val depth : 'a tree -> int = <fun>
val preorder : 'a tree -> 'a list = <fun>

Why did we pick fold_right and not fold_left for this development? Because fold_left is tail recursive, which is something we’re never going to achieve on binary trees. Suppose we process the left branch first; then we still have to process the right branch before we can return. So there will always be work left to do after a recursive call on one branch. Thus on trees an equivalent to fold_right is the best which we can hope for.

The technique we used to derive fold_tree works for any OCaml variant type t:

  • Write a recursive fold function that takes in one argument for each constructor of t.

  • That fold function matches against the constructors, calling itself recursively on any value of type t that it encounters.

  • Use the appropriate argument of fold to combine the results of all recursive calls as well as all data not of type t at each constructor.

This technique constructs something called a catamorphism, aka a generalized fold operation. To learn more about catamorphisms, take a course on category theory.

4.5.3. Filter on Trees

This one is perhaps the hardest to design. The problem is: if we decide to filter a node, what should we do with its children?

  • We could recurse on the children. If after filtering them only one child remains, we could promote it in place of its parent. But what if both children remain, or neither? Then we’d somehow have to reshape the tree. Without knowing more about how the tree is intended to be used—that is, what kind of data it represents—we are stuck.

  • Instead, we could just eliminate the children entirely. So the decision to filter a node means pruning the entire subtree rooted at that node.

The latter is easy to implement:

let rec filter_tree p = function
  | Leaf -> Leaf
  | Node (v, l, r) ->
    if p v then Node (v, filter_tree p l, filter_tree p r) else Leaf
val filter_tree : ('a -> bool) -> 'a tree -> 'a tree = <fun>