-How would we move upward in a tree? Well, we'd build a regular, untargetted tree with a root node---let's call it `20`---and whose children are given by the outermost sibling list in the targetted tree above, after inserting the targetted subtree into the `*` position:
-
- node 20
- / | \
- / | \
- leaf 1 leaf 2 leaf 3
-
-We'll call this new untargetted tree `node 20`. The result of moving upward from our previous targetter tree, targetted on `leaf 1`, would be the outermost `parent` element of that targetted tree, with `node 20` being the subtree that fills that parent's target position `*`:
-
- {
- parent = ...;
- siblings = [*; node 50; node 80]
- }, * filled by node 20
-
-Or, spelling that structure out fully:
-
- {
- parent = {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; node 920; node 950]
- },
- siblings = [*; node 50; node 80]
- }, * filled by node 20
-
-Moving upwards yet again would get us:
-
- {
- parent = {
- parent = None;
- siblings = [*]
- },
- siblings = [*; node 920; node 950]
- }, * filled by node 500
-
-where `node 500` refers to a tree built from a root node whose children are given by the list `[*; node 50; node 80]`, with `node 20` inserted into the `*` position. Moving upwards yet again would get us:
-
- {
- parent = None;
- siblings = [*]
- }, * filled by node 1000
-
-where the targetted element is the root of our base tree. Like the "moving backward" operation for the list zipper, this "moving upward" operation is supposed to be reminiscent of closing a zipper, and that's why these data structures are called zippers.
-
-We haven't given you a real implementation of the tree zipper, but only a suggestive notation. We have however told you enough that you should be able to implement it yourself. Or if you're lazy, you can read:
-
-* [[!wikipedia Zipper (data structure)]]
-* Huet, Gerard. ["Functional Pearl: The Zipper"](http://www.st.cs.uni-sb.de/edu/seminare/2005/advanced-fp/docs/huet-zipper.pdf) Journal of Functional Programming 7 (5): 549-554, September 1997.
-* As always, [Oleg](http://okmij.org/ftp/continuations/Continuations.html#zipper) takes this a few steps deeper.
-
-##Same-fringe using a tree zipper##
-
-Supposing you did work out an implementation of the tree zipper, then one way to determine whether two trees have the same fringe would be: go downwards (and leftwards) in each tree as far as possible. Compare the targetted leaves. If they're different, stop because the trees have different fringes. If they're the same, then for each tree, move rightward if possible; if it's not (because you're at the rightmost position in a sibling list), more upwards then try again to move rightwards. Repeat until you are able to move rightwards. Once you do move rightwards, go downwards (and leftwards) as far as possible. Then you'll be targetted on the next leaf in the tree's fringe. The operations it takes to get to "the next leaf" may be different for the two trees. For example, in these trees:
-
- . .
- / \ / \
- . 3 1 .
- / \ / \
- 1 2 2 3
-
-you won't move upwards at the same steps. Keep comparing "the next leafs" until they are different, or you exhaust the leafs of only one of the trees (then again the trees have different fringes), or you exhaust the leafs of both trees at the same time, without having found leafs with different labels. In this last case, the trees have the same fringe.
-
-If your trees are very big---say, millions of leaves---you can imagine how this would be quicker and more memory-efficient than traversing each tree to construct a list of its fringe, and then comparing the two lists so built to see if they're equal. For one thing, the zipper method can abort early if the fringes diverge early, without needing to traverse or built a list containing the rest of each tree's fringe.
-
-Let's sketch the implementation of this. We won't provide all the details for an implementation of the tree zipper, but we will sketch an interface for it.
-
-First, we define a type for leaf-labeled, binary trees:
-
- type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
-
-Next, the interface for our tree zippers. We'll help ourselves to OCaml's **record types**. These are nothing more than tuples with a pretty interface. Instead of saying:
-
- # type blah = Blah of (int * int * (char -> bool));;
-
-and then having to remember which element in the triple was which:
-
- # let b1 = Blah (1, (fun c -> c = 'M'), 2);;
- Error: This expression has type int * (char -> bool) * int
- but an expression was expected of type int * int * (char -> bool)
- # (* damnit *)
- # let b1 = Blah (1, 2, (fun c -> c = 'M'));;
- val b1 : blah = Blah (1, 2, <fun>)
-
-records let you attach descriptive labels to the components of the tuple:
-
- # type blah_record = { height : int; weight : int; char_tester : char -> bool };;
- # let b2 = { height = 1; weight = 2; char_tester = fun c -> c = 'M' };;
- val b2 : blah_record = {height = 1; weight = 2; char_tester = <fun>}
-
-These were the strategies to extract the components of an unlabeled tuple:
-
- let h = fst some_pair;; (* accessor functions fst and snd are only predefined for pairs *)
-
- let (h, w, test) = b1;; (* works for arbitrary tuples *)
-
- match b1 with
- | (h, w, test) -> ...;; (* same as preceding *)
-
-Here is how you can extract the components of a labeled record:
-
- let h = b2.height;; (* handy! *)
-
- let {height = h; weight = w; char_tester = test} = b2
- in (* go on to use h, w, and test ... *)
-
- match test with
- | {height = h; weight = w; char_tester = test} ->
- (* go on to use h, w, and test ... *)
-
-Anyway, using record types, we might define the tree zipper interface like so:
-
- type 'a starred_tree = Root | Starring_Left of 'a starred_pair | Starring_Right of 'a starred_pair
- and 'a starred_pair = { parent : 'a starred_tree; sibling: 'a tree }
- and 'a zipper = { tree : 'a starred_tree; filler: 'a tree };;
-
- let rec move_botleft (z : 'a zipper) : 'a zipper =
- (* returns z if the targetted node in z has no children *)
- (* else returns move_botleft (zipper which results from moving down and left in z) *)
-
-<!--
- let {tree; filler} = z
- in match filler with
- | Leaf _ -> z
- | Node(left, right) ->
- let zdown = {tree = Starring_Left {parent = tree; sibling = right}; filler = left}
- in move_botleft zdown
- ;;
--->
-
- let rec move_right_or_up (z : 'a zipper) : 'a zipper option =
- (* if it's possible to move right in z, returns Some (the result of doing so) *)
- (* else if it's not possible to move any further up in z, returns None *)
- (* else returns move_right_or_up (result of moving up in z) *)
-
-<!--
- let {tree; filler} = z
- in match tree with
- | Starring_Left {parent; sibling = right} -> Some {tree = Starring_Right {parent; sibling = filler}; filler = right}
- | Root -> None
- | Starring_Right {parent; sibling = left} ->
- let z' = {tree = parent; filler = Node(left, filler)}
- in move_right_or_up z'
- ;;
--->
-
-The following function takes an 'a tree and returns an 'a zipper focused on its root:
-
- let new_zipper (t : 'a tree) : 'a zipper =
- {tree = Root; filler = t}
- ;;
-
-Finally, we can use a mutable reference cell to define a function that enumerates a tree's fringe until it's exhausted:
-
- let make_fringe_enumerator (t: 'a tree) =
- (* create a zipper targetting the root of t *)
- let zstart = new_zipper t
- in let zbotleft = move_botleft zstart
- (* create a refcell initially pointing to zbotleft *)
- in let zcell = ref (Some zbotleft)
- (* construct the next_leaf function *)
- in let next_leaf () : 'a option =
- match !zcell with
- | None -> (* we've finished enumerating the fringe *)
- None
- | Some z -> (
- (* extract label of currently-targetted leaf *)
- let Leaf current = z.filler
- (* update zcell to point to next leaf, if there is one *)
- in let () = zcell := match move_right_or_up z with
- | None -> None
- | Some z' -> Some (move_botleft z')
- (* return saved label *)
- in Some current
- )
- (* return the next_leaf function *)
- in next_leaf
- ;;
-
-Here's an example of `make_fringe_enumerator` in action:
-
- # let tree1 = Leaf 1;;
- val tree1 : int tree = Leaf 1
- # let next1 = make_fringe_enumerator tree1;;
- val next1 : unit -> int option = <fun>
- # next1 ();;
- - : int option = Some 1
- # next1 ();;
- - : int option = None
- # next1 ();;
- - : int option = None
- # let tree2 = Node (Node (Leaf 1, Leaf 2), Leaf 3);;
- val tree2 : int tree = Node (Node (Leaf 1, Leaf 2), Leaf 3)
- # let next2 = make_fringe_enumerator tree2;;
- val next2 : unit -> int option = <fun>
- # next2 ();;
- - : int option = Some 1
- # next2 ();;
- - : int option = Some 2
- # next2 ();;
- - : int option = Some 3
- # next2 ();;
- - : int option = None
- # next2 ();;
- - : int option = None
-
-You might think of it like this: `make_fringe_enumerator` returns a little subprogram that will keep returning the next leaf in a tree's fringe, in the form `Some ...`, until it gets to the end of the fringe. After that, it will keep returning `None`.
-
-Using these fringe enumerators, we can write our `same_fringe` function like this:
-
- let same_fringe (t1 : 'a tree) (t2 : 'a tree) : bool =
- let next1 = make_fringe_enumerator t1
- in let next2 = make_fringe_enumerator t2
- in let rec loop () : bool =
- match next1 (), next2 () with
- | Some a, Some b when a = b -> loop ()
- | None, None -> true
- | _ -> false
- in loop ()
- ;;
-
-The auxiliary `loop` function will keep calling itself recursively until a difference in the fringes has manifested itself---either because one fringe is exhausted before the other, or because the next leaves in the two fringes have different labels. If we get to the end of both fringes at the same time (`next1 (), next2 ()` matches the pattern `None, None`) then we've established that the trees do have the same fringe.