## Description

Be sure to include your name at the beginning of each ﬁle! Assignment 8 include a programming portion and a written part. The programming portion must compile and consist of a single ﬁle ( hw08.cpp). The written portion should consist of a single ﬁle (hw08written) in a .pdf format. Be sure to include your name at the beginning of each ﬁle! You must hand in the ﬁle via NYU Classes. You must hand in both ﬁles via NYU Classes.

Programming Part

1. In this part, you will store a list of correctly spelled words and a point value associated with each word into a variable of type map

On your own1. You can write a program to help you play Words With Friends. Remember the recursive function from the extra credit problem in a previous assignment, where the user enters a string and you ﬁnd all the combinations of the string. For each string created from the recursive function, you can test to see if it is in the ENABLE word list. If so, you print out the word and the points associated with the word. Use the map

∗10% extra credit will be given if you turn this assignment in on Tuesday April 12, 2016 at 11:00 p.m. 1Do not turn this in

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2. Add the following methods to the BinarySearchTree class.

(a) Implement the method contains recursively. (b) Implement a method that takes two keys, low and high, and prints all the objects X that are in the range speciﬁed by low and high. (i.e. all keys in the range [low, high].) Your program should run in O(k + h) time, where k is the number of keys printed and h is the height of the tree. Thus if k is small, you should be examining only a small part of the tree. Use a hidden recursive method and do not use an inorder traversal. Bound the running time of your algorithm using Big-Oh notation. (c) Ever feel contrarian? So much eﬀort is spent creating a balanced BST. In this problem2 you are to write a method called stringy that creates an unbalanced BST, a ”stringy” BST, from a regular BST. If your tree contained

d / \ b f / \ / a c e after running stringy, all nodes in your tree should have their left member variable contain the nullptr. Thus the tree should have height n−1. In the example above, after calling stringy the tree would look like:

a

\ b

\

c

\

d

\

e

\

f

Your method must not create any new nodes3. I have removed the size method from the node class, so you do not need to update the size of each node.

(d) Create a method called average_node_depth that computes the average depth of a node in the tree. e.g.

6 / \ 3 8 / \ 2 4 \ 5 Then the average depth of a node is (0+1+2+2+3+1)/6 = 9/6 since there is one node at depth 1, two nodes at depth 2, one node at depth 3.4 2This problem was suggested by Shahzaib. 3Please take the time to ﬁgure out how to solve this method on paper (using pictures) before you start coding. 4The average number of comparisons to ﬁnd a item in this tree is 9/6+1.

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3. Add a method to the binary search tree class that lists the nodes of a binary tree in level-order (It should ﬁrst list the root, then the nodes at depth 1, then the nodes at depth 2, and so on). Your algorithm should have a worst case running time of O(n), where n is the number of nodes in the tree. Explain how your algorithm fulﬁlls this requirement. Hint: a Queue

e.g. For the following tree:

31 / \ 12 53 / \ / \ 4 25 46 67

Printing the nodes of this tree in a level order would output: 31,12,53,4,25,46,67.

4. (Extra Credit) You may do two of the following three extra extra credit problems. The points associated with each problem with be posted on Piazza.

(a) Run empirical studies to estimate the average height of a node in a binary search tree by running 100 trial of inserting n random keys into an initially empty tree. For n = 210,211,212. For each n print the min, the max, and the average. (b) Add6 iterators to the binary search tree class. To do this: i. add two extra pointers to the node class. Use these pointers to link each node to the next smallest and next largest node. ii. add a header and tail node which are not part of the binary search tree. Adding these nodes helps to make the code simpler iii. add the methods begin( ) and end( ) to the binary search tree class (c) See7 Shah’s Memory.pdf.

5Level-order traversal of a tree is also called breadth-ﬁrst traversal. I suggest you do not use recursion 6This problem was suggested by Shahzaib. 7This problem is from Shahzaib.

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Written Part

1. Write the pseudo code for: • programming problem 2. For each method, write the 3 to 8 steps needed to solve this problem. 2. For the programming problems in question 2, determine the running times of your implementations using Big-Oh notation.

3. This code is modiﬁed from the code we discussed in class. Does this code perform correctly? If not, describe all problems of this code and ﬁx the code.

template

return t;

}

4. This code is modiﬁed from the code we discussed in class. Does this code perform correctly? If not, describe all problems of this code and ﬁx the code.

void insert( const Comparable & x, Node * t ) { if( t == nullptr ) t = new Node{ x, nullptr, nullptr, 1 }; else if( x < t->element ) insert( x, t->left ); else if( t->element < x ) insert( x, t->right ); else ; // Duplicate; do nothing t->size++; }

5. For the following tree:

22

17

14 18

30

(a) what is the output of an inorder traversal of the tree? (b) what is the output of an preorder traversal of the tree? (c) what is the output of an postorder traversal of the tree?