More Binary Search Trees

Goals

This lab provides more experience with binary search trees, building on the previous lab on trees.

Background

1. Work in this lab involves the following classes that were introduced in the previous lab.

   • Person
   • Student
   • Faculty
   • TreeNode
   • BSTree
   • DirectoryBST

   Be sure that these classes are available within the eclipse environment and available for further development.

Computing the Height of a Tree

The reading for this lab defines the length of a path within a tree, the depth or level of a node within a tree, and the height of a tree.

2. Consider the following binary search tree:

   

   1. Identify the levels of the nodes with labels a, f, g, k, l, and p.
   2. What is the height of this tree? Briefly explain your answer.

3. Add a method nodeLevel to BSTree that returns the level of a node; nodeLevel should take a node's data (class E) as parameter. (If the tree does not contain a node with the given data, the method should return -1.)

4. Write a method height that returns the height of a binary search tree.
   (Use of global variables for this method will be subject to a substantial penalty — perhaps 75%.)

Determining if a Tree is Height Balanced

Today's reading draws upon Computer Science 2: Principles of Software Engineering, Data Types, and Algorithms by Henry M. Walker [Scott, Foresman and Company, 1989]. In particular, the reading defines when a tree is balanced or completely balanced.

5. Consider tree B above, and consider tree A below:

   

   Determine if either of these trees is balanced. In each case, justify your answer.

6. Write a method isBalanced that determines if a given binary search tree is (completely) balanced.
   (Use of global variables for this method will be subject to a substantial penalty — perhaps 75%.)

Today's reading also defines when a tree is height-balanced.

7. Apply this definition to determine if the trees in Tree C and Tree D (below) are height balanced.

   
8. Write a method isHeightBalanced that determines if a given binary search tree is height balanced.
   (Use of global variables for this method will be subject to a substantial penalty — perhaps 75%.)

Deletion of a Node

Removal of a node with both left and right children

In the original Tree B, suppose we wish to remove the root, node k. We cannot just remove node k, since it connects two subtrees. Also, we cannot just move the left or right subtree upward to the root, because we must include the other subtree in the final structure as well. Instead, we first consider what values might replace 'k' in the root. In order to retain the ordering of nodes within a search tree, there are only two choices:

• we could promote 'j': the largest value in the left subtree, or
• we could promote 'l': the smallest value in the right subtree.

9. Write a brief argument explaining why these are the only two values that could be moved to the root.

The remove Method

10. Write a method remove that takes a value as parameter and that removes the node with that value from the binary search tree.
    (Use of global variables for this method will be subject to a substantial penalty — perhaps 75%.)

Tree Traversals

11. Consider a binary search tree of Person objects, as defined at the start of this lab and in the reading. Write a method

          Person findLongest (BSTree<Person>)
        

    that searches the tree for the Person with the longest first name (i.e., the first name of the Person has the most number of letters). In case the first names in two or more Persons have the same number of letters, findLongest should return the Person for which the first name comes last in dictionary order.
    (As with other steps in this labs, the use of global variables for this method wil be subject to a substantial penalty — up to 75%.)

12. For your findLongest method in step 11, suppose the initial tree contains n nodes, and suppose the height of the overall tree is k. Analyze your code to determine the order O(???) of the algorithm— giving both the careful analysis of work done and a clear statement of the conclusion.

created 21 April 2000 revised 24 March 2005 revised 24 April 2012 revised 3 November 2018 minor editing 31 January 2020
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.