midterm1-sp12-blank

# Let t1 int tree node node node empty 1 empty 2 empty

• Notes
• 11

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let t1 : int tree = Node (Node (Node (Empty, 1, Empty), 2, Empty), 3, Empty) (a) (b) (c) (d) (e) none of the above let t2 : int tree = Node (Empty, 3, Node (Empty, 2, Node (Empty, 1, Empty))) (a) (b) (c) (d) (e) none of the above let t3 : int tree = Node (Empty, 1, Node (Empty, 2, Node (Empty, 3, Empty))) (a) (b) (c) (d) (e) none of the above let t4 : int tree = Node (Node (Empty, 1, Empty), 2, Node (Empty, 3, Empty)) (a) (b) (c) (d) (e) none of the above 7

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c. (12 points) Complete this definition of a function that returns the leaves of the given tree from left-to-right. For example, calling leaves on tree (a) returns [1;0;4] . You may use the @ operator (i.e. list append) in your solution. let rec leaves (t:’a tree) : ______________ = 8
5. Binary Search Trees (21 points) a. (9 points) Recall the delete function for binary search trees from class. (This function uses the same tree datatype from the previous problem.) let rec tree_max (t:’a tree) : ’a = begin match t with | Empty -> failwith "tree_max called on empty tree" | Node(_,x,Empty) -> x | Node(_,_,rt) -> tree_max rt end let rec delete (t:’a tree) (n:’a) : ’a tree = begin match t with | Empty -> Empty | Node(lt,x,rt) -> if x = n then begin match (lt, rt) with | (Empty, Empty) -> Empty | (Empty, rt) -> rt | (lt, Empty) -> lt | (lt, rt) -> let y = tree_max lt in (Node (delete lt y, y, rt)) end else if n < x then Node(delete lt n, x, rt) else Node(lt, x, delete rt n) end (This problem continues on the next page.) 9

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Let t be the BST depicted below. 6 / \ 4 7 / \ \ 2 5 8 For each separate call to delete with this tree , draw the result: delete t 2 delete t 7 delete t 6 10
b. (12 points) Implement bst_filter . The bst_filter function applies a given predicate to each element in an input tree to see if it should be included in the output. (This function is analogous to the list filter function from homework four.) For example below, filtering the tree on the left with a predicate for even numbers results in the tree on the right: 6 6 / \ bst_filter is_even / \ 4 7 --------> 4 8 / \ \ / 2 5 8 2 Below, complete the definition, including the types of pred and the result type of the func- tion. In your implementation, you must use the BST delete function. let rec bst_filter (pred: ________________) (t : ’a tree) : ____________ = 11
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• Fall '09
• Self-balancing binary search tree, val empty, int list list

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