Assignment -
Question 1: Property of Binary Trees
Assume a proper and complete binary tree (T) with n > 1 nodes. Let E(T) represent the sum of the depths of all external nodes in T (see Figure1), and I(T) represent the sum of the depths of all internal nodes in T. Prove that:
(a) E(T) = O(n⋅log(n))
(b) E(T) - I(T) = 2i, where i represents the actual number of internal nodes in T.
Question 2: Binary Search Tree
Given a BST and two numbers - a and b (where a ≤ b) - propose an algorithm that prints all the elements k of the BST that satisfy: k ≥ a and k ≤ b. Your algorithm should return the found keys in ascending order. You can assume that there are no duplicate keys in the tree. (You can also assume that a, b, and all the keys stored in the BST are integers.)
In addition to writing a brief description of the algorithm, you also need to implement the algorithm in Java (i.e., provide respective Java code), as well as justify the algorithm's running time.
Question 3: BST-based Text Analyzer
In this project, you are required to write a program that builds a binary search tree (BST) of distinct words from an input text file. Each time a new word occurs in the text, it is inserted into the tree; if the word has previously occurred, a count associated with the word is incremented. (Note: you will have to parse the text obtained from the file to find the words before inserting them into the tree. Furthermore, you will have to replace all capital letters with their lower-case equivalents and remove punctuation.)
Once all words are inserted into the BST, the following three experiments should be performed.
1) Compute the maximum length of all search paths in the BST.
2) Print all distinct words found in the text (in sorted/alphabetic order).
3) Find the ten most common words in the text. You should output both the word and the number of times it occurred.
Your program design should be based on the following guidelines:
(1) Create BTNode1 class, which will be used to store each distinct word together with is occurrence-counter (wordCounter). The outline of this class is given below.
(2) Create LinkedBinaryTree class as outlined in the textbook and class-notes.
(3) Create BinarySearchTree class (extends LinkedBinaryTree), which will be used to store all distinct words form a given file. The outline of this class is given below. You are allowed to add new variables and methods to this class, as needed.
(4) Create TextAnalyzer class, which will contain main() method and act as a tester class. The outline of this class is given below. You are allowed to add new methods to this class, as needed.
Attachment:- Assignment Files.rar