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. Show us some of your tests. type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree) type 'a starred_level = Root | Starring_Left of 'a starred_nonroot | Starring_Right of 'a starred_nonroot and 'a starred_nonroot = { parent : 'a starred_level; sibling: 'a tree };; type 'a zipper = { level : 'a starred_level; 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 from z to the leftmost child) *) _____ (* 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 = {level = Root; filler = t} ;;   let make_fringe_enumerator (t: 'a tree) = (* create a zipper targetting the botleft of t *) let zbotleft = move_botleft (new_zipper t) (* 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 | 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 | None -> (* we've finished enumerating the fringe *) None ) (* 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 (`tailk`s) as arguments. Your assignment is to fill in the blanks in the code, **and also to supply comments to the code,** to explain what every significant piece is doing. Don't forget to supply the comments, this is an important part of the assignment. 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) ; then result1 ; else (if (test2 argument argument) ; then result2 ; else (if (test3 argument argument) ; then result3 ; else 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 as the preceding * `'(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 _____ tailk)))] [else (cons tree tailk)]))]) (helper tree (lambda () _____)))) (define (stream-equal? stream1 stream2) (cond [(and (null? stream1) (null? stream2)) _____] [(and (pair? stream1) (pair? stream2)) (and (equal? (car stream1) (car 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)