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1 CSCA48 Winter 2017 Term Test #2 Duration 80 minutes Aids allowed: none Last Name: Student Number: Markus Login: First Name: Question 0. [1 mark] Carefully read and follow all instructions on this page, and fill in all fields. Please place a checkmark ( ) beside your tutorial session Tutorial Number Date/Time Room TA Name Check TUT0001 MO 09:00 10:00 IC 208 Eric Wang TUT0002 MO 10:00 11:00 IC 208 Prantar TUT0003 MO 11:00 12:00 AA 204 Vincent Cuenca TUT0004 MO 12:00 13:00 IC 326 Ralph TUT0005 FR 12:00 13:00 HW 214 Faizan TUT0006 MO 16:00 17:00 AC 334 Andrew Wang TUT0007 TU 09:00 10:00 IC 320 Jana TUT0008 TU 10:00 11:00 IC 326 Brian TUT0009 TU 11:00 12:00 IC 326 Kiwi TUT0010 TU 12:00 13:00 IC 326 Bilal TUT0013 WE 10:00 11:00 IC 326 Faizan TUT0014 WE 12:00 13:00 IC 120 Harman TUT0015 TU 15:00 16:00 IC 326 Aiyaz TUT0019 FR 14:00 15:00 MW 110 Lev TUT0020 WE 10:00 11:00 BV 355 Ralph TUT0021 FR 12:00 13:00 BV 363 David Liu TUT0022 WE 16:00 17:00 HW 215 Vincent Tse TUT0023 TH 11:00 12:00 IC 320 Bilal TUT0024 TH 12:00 13:00 IC 320 Mahima TUT0026 WE 13:00 14:00 BV 355 Maria Sirilan Do not turn this page until you have received the signal to start. This exam consists of 3 questions on 16 pages (including this one). When you receive the signal to start, please make sure that your copy is complete. Proper documentation is required for all functions and code blocks. If you use any space for rough work, indicate clearly what you want marked. Please read all questions thoroughly before starting on any work. The University of Toronto s Code of Behaviour on Academic Matters applies to all University of Toronto Scarborough students. The Code prohibits all forms of academic dishonesty including, but not limited to, cheating, plagiarism, and the use of unauthorized aids. Students violating the Code may be subject to penalties up to and including suspension or expulsion from the University. # 0: / 1 # 1: /10 # 2: /10 # 3: /12 TOTAL: /33 Total Pages = 16 Page 1 cont d...

2 [Use the space below for rough work. This page will not be marked unless you clearly indicate the part of your work that you want us to mark.] Student #: Page 2 of 16 cont d...

3 Question 1. Part (a) [2 marks] [10 marks] How many ways can we order the insertions of the numbers 1, 2, 3, 4 into an initially empty binary search tree so that the resultant tree would have height 3? No justification is required. Put another way, there are 4! = 24 ways to order the insertions of the numbers 1, 2, 3, 4, and we want to know how many of them would yield a tree of height 3. Part (b) [4 marks] For n > 0, how many ways can we order the insertions of the numbers 1,, n into an initially empty binary search tree so that the resultant tree would have height n 1? Justify your answer. You will receive no credit without correct justification. Part (c) [4 marks] How many ways can a (min) heap contain exactly the letters O, U, T, S, I, D, E, R, and all the vowels are in the left subtree of the root of the heap? Justify your answer. You will receive no credit without correct justification. You must use the implementation of the heap as shown in lecture the one that uses a list to store a compact binary tree. In case you need it, here are the letters O, U, T, S, I, D, E, R in alphabetical order: D, E, I, O, R, S, T, U. Also, the vowels are E, I, O, U. Student #: Page 3 of 16 cont d...

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5 Question 2. [10 marks] Consider the following recursive function. def jumble(s : str) -> str: if len(s) < 2: result = s else: s = s[1:] + s[0] # move first symbol to end h = len(s) // 2 # integer division, round down t = jumble(s[h:], indent + " ") # jumble second half u = jumble(s[:h], indent + " ") # jumble first half result = t + u # jumbled second half + jumbled first half return result Part (a) [4 marks] Using the convention of tracing as shown in lecture, trace jumble("pineapple"). Student #: Page 5 of 16 cont d...

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7 Part (b) [2 marks] Using as many different upper case letters as possible, give a string s such that len(s)==4 and jumble(s)==s. No justification is required. Examples to illustrate different upper case letters : "MORE" contains 4 different upper case letters. "LESS" contains 3 different upper case letters. "NAAN" contains 2 different upper case letters. "ZZZZ" contains 1 different upper case letter. Part (c) [2 marks] Using as many different upper case letters as possible, give a string t such that len(t)==5 and jumble(t)==t. No justification is required. Part (d) [2 marks] Using as many different upper case letters as possible, give a string u such that len(u)==6 and jumble(u)==u. No justification is required. For your convenience, here is the same function again. def jumble(s : str) -> str: if len(s) < 2: result = s else: s = s[1:] + s[0] # move first symbol to end h = len(s) // 2 # integer division, round down t = jumble(s[h:], indent + " ") # jumble second half u = jumble(s[:h], indent + " ") # jumble first half result = t + u # jumbled second half + jumbled first half return result Student #: Page 7 of 16 cont d...

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9 Question 3. [12 marks] For use as an example in class, Nick wrote some beautiful code that solves an interesting tree problem using a recursive helper function. Then disaster struck! First of all the Code Stealer came and erased the entire body of the function unright avg. Secondly the Comment Thief came and erased the type contract and the description of the function unright helper. Thirdly the Variable Anonymizer came and changed some variable names to names that are cryptic and nondescript. Names that have been changed begin with VAR1 or VAR2 (VAR stands for Variable Anonymizer Rule). Finally the Code Mangler came and mangled the body of the function unright helper by (i) removing all indentation, and (ii) shuffling all the lines of code. Please help Nick restore his code below. Lines with ### indicate where restoration is needed. Advice: Read the last function (unright avg) first to understand what the code is all about. Here is the class for the nodes. No restoration is needed here. class IntBinTreeNode: """ A node containing an integer in a binary tree. """ def init (self, num, left, right): """ (BinTreeNode, int, IntBinTreeNode, IntBinTreeNode) -> NoneType Initialize a new IntBinTreeNode with num, left and right children. """ self.num = num # the integer stored in the node self.left = left # pointer to left child/subtree self.right = right # pointer to right child/subtree Student #: Page 9 of 16 cont d...

10 [Use the space below for rough work. This page will not be marked unless you clearly indicate the part of your work that you want us to mark.] Student #: Page 10 of 16 cont d...

11 Here is the docstring for the helper function. def unright_helper(root, include_root): """ ### Restore type contract and function description here. >>> root = IntBinTreeNode(1,\ IntBinTreeNode(2,\ IntBinTreeNode(3, None, None),\ IntBinTreeNode(4, None, None)),\ IntBinTreeNode(5,\ IntBinTreeNode(6, None, None),\ IntBinTreeNode(7, None, None))) >>> unright_helper(root, True) (12, 4) >>> unright_helper(root, False) (11, 3) >>> unright_helper(none, True) (0, 0) >>> unright_helper(none, False) (0, 0) """ Student #: Page 11 of 16 cont d...

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13 Here is the body of the helper function. ### Give appropriate names for VAR1 and VAR2, and unmangle the code. ### Here is the mangled code, with some bad variable names. ### (VAR1_left, VAR2_left) = unright_helper(root.left, True) ### (VAR1_right, VAR2_right) = unright_helper(root.right, False) ### VAR1 = 0 ### VAR1 = VAR1 + root.num ### VAR1 = VAR1_left + VAR1_right ### VAR2 = 0 ### VAR2 = VAR2 + 1 ### VAR2 = VAR2_left + VAR2_right ### if include_root: ### if root!= None: ### return (VAR1, VAR2) Student #: Page 13 of 16 cont d...

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15 Here is the function that calls the helper function. def unright_avg(root): """ (IntBinTreeNode) -> float Return the average of all the numbers in the tree rooted at root that are not stored in a node that is a right child (of another node). I.e., take all the nodes that are not right children (including root), and return the average of the numbers stored in them. >>> root = IntBinTreeNode(1,\ IntBinTreeNode(2,\ IntBinTreeNode(3, None, None),\ IntBinTreeNode(4, None, None)),\ IntBinTreeNode(5,\ IntBinTreeNode(6, None, None),\ IntBinTreeNode(7, None, None))) >>> unright_avg(root) 3.0 REQ: root!= None. I.e., tree is not empty. """ ### Restore body of this function. Student #: Page 15 of 16 cont d...

16 [Use the space below for rough work. This page will not be marked unless you clearly indicate the part of your work that you want us to mark.] Page 16 of 16 End of Examination