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midterm1-fall12-solutions

# Some of these definitions refer to functions from the

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Unformatted text preview: Some of these definitions refer to functions from the Map1 module, which has the following abstract interface: module type MAP = sig type (’a, ’b) map val empty : (’a, ’b) map val is_empty : (’a, ’b) map -> bool val mem : ’a -> (’a, ’b) map -> bool val find : ’a -> (’a, ’b) map -> ’b val add : ’a -> ’b -> (’a, ’b) map -> (’a, ’b) map val remove : ’a -> (’a, ’b) map -> (’a, ’b) map val from_list : (’a * ’b) list -> (’a, ’b) map val bindings : (’a, ’b) map -> (’a * ’b) list end module Map1 : MAP = struct ... end ;; open Map1 let x : ______ (int,string) map _____________ = add 120 "is fun" empty let a : _______int list list ________________ = (2::):: let b : ________ill typed____________________ = 2 + "three" let c : ________(int, bool) map______________ = add 3 true empty let d : ___(int,bool) map -> (int, bool) map_ = add 3 true let e : ____ill typed________________________ = mem 3 [1;2;3] let f : ____(int -> int) -> int______________ = fun (g:int -> int) -> g 3 let g : _____int -> int -> int_______________ = fun (x:int) (y:int) -> x + y let h : _____(int,(int,int) map) map_________ = add 3 (from_list [(1,2)]) empty Grading scheme: 2 points per answer: 0 if wrong, 2 if right. 6 4. Binary Search Trees (17 points) Recall the definition of generic binary trees and the binary search tree insert function: type ’a tree = | Empty | Node of ’a tree * ’a * ’a tree let rec insert (t:’a tree) (n:’a) : ’a tree = begin match t with | Empty -> Node(Empty, n, Empty) | Node(lt, x, rt) -> if x = n then t else if n < x then Node (insert lt n, x, rt) else Node(lt, x, insert rt n) end a. (5 points) Circle the trees that satisfy the binary search tree invariant . (Note that we have omitted the Empty nodes from these pictures.) (a) (b) (c) (d) (e) 4 2 2 2 2 / \ / \ / \ \ \ 2 5 5 6 5 6 5 5 \ / \ / \ / \ 6 4 4 4 6 4 4 Answer: (a), (d) b. (12 points) For each definition below, circle the letter of the tree above that it constructs or “none of the above”. let t1 : int tree = insert (Node(Node(Empty, 5 Empty), 2, Node(Empty, 6, Empty))) 4 (a) (b) (c) (d) (e) none of the above Answer: (b) let t2 : int tree = insert (insert (insert (insert Empty 4) 2) 5) 6 (a) (b) (c) (d) (e) none of the above Answer: (a) let t3 : int tree = insert (insert (insert (insert Empty 2) 5) 4) 6 (a) (b) (c) (d) (e) none of the above Answer: (d) 7 let t4 : int tree = insert (insert (insert (insert Empty 5) 2) 4) 6 (a) (b) (c) (d) (e) none of the above Answer: none of the above 8 5. Lists and Binary Trees (20 points)5....
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