1. Complete the definitions of `move_botleft` and `move_right_or_up` from the same-fringe solution in the [[week11]] notes. Test your attempts against some example trees to see if the resulting `make_fringe_enumerator` and `same_fringe` functions work as expected.
type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree)
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) *)
YOU SUPPLY THE DEFINITION
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) *)
YOU SUPPLY THE DEFINITION
let new_zipper (t : 'a tree) : 'a zipper =
{tree = Root; filler = t}
;;
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
;;
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 ()
;;
2. Here's another implementation of the same-fringe function, in Scheme. It's taken from . It uses thunks to delay the evaluation of code that computes the tail of a list of a tree's fringe. It also involves passing continuations as arguments. Your assignment is to supply comments to the code, to explain what every significant piece is doing.
This code uses Scheme's `cond` construct. That works like this;
(cond
((test1 argument argument) result1)
((test2 argument argument) result2)
((test3 argument argument) result3)
(else result4))
is equivalent to:
(if (test1 argument argument)
result1
(if (test2 argument argument)
result2
(if (test3 argument argument)
result3
result4)))
Some other Scheme details:
* `#t` is true and `#f` is false
* `(lambda () ...)` constructs a thunk
* there is no difference in meaning between `[...]` and `(...)`; we just sometimes use the square brackets for clarity
* `'(1 . 2)` and `(cons 1 2)` are pairs (the same pair)
* `(list)` and `'()` both evaluate to the empty list
* `(null? lst)` tests whether `lst` is the empty list
* non-empty lists are implemented as pairs whose second member is a list
* `'()` `'(1)` `'(1 2)` `'(1 2 3)` are all lists
* `(list)` `(list 1)` `(list 1 2)` `(list 1 2 3)` are the same lists
* `'(1 2 3)` and `(cons 1 '(2 3))` are both pairs and lists (the same list)
* `(pair? lst)` tests whether `lst` is a pair; if `lst` is a non-empty list, it will also pass this test; if `lst` fails this test, it may be because `lst` is the empty list, or because it's not a list or pair at all
* `(car lst)` extracts the first member of a pair / head of a list
* `(cdr lst)` extracts the second member of a pair / tail of a list
Here is the implementation:
(define (lazy-flatten tree)
(letrec ([helper (lambda (tree tailk)
(cond
[(pair? tree)
(helper (car tree) (lambda () (helper (cdr tree) tailk)))]
[else (cons tree tailk)]))])
(helper tree (lambda () (list)))))
(define (stream-equal? stream1 stream2)
(cond
[(and (null? stream1) (null? stream2)) #t]
[(and (pair? stream1) (pair? stream2))
(and (equal? (car stream1) (car stream2))
(stream-equal? ((cdr stream1)) ((cdr stream2))))]
[else #f]))
(define (same-fringe? tree1 tree2)
(stream-equal? (lazy-flatten tree1) (lazy-flatten tree2)))
(define tree1 '(((1 . 2) . (3 . 4)) . (5 . 6)))
(define tree2 '(1 . (((2 . 3) . (4 . 5)) . 6)))
(same-fringe? tree1 tree2)