BINARY SEARCH TREES

BST basics. Know the definition of a BST and how to search and insert.

Recursive BST code. You should be able to read recursive BST code. You will not be required to write recursive BST code, since you won't have practice until the kd-tree assignment.

In-order traversal. Understand how to perform an inorder traversal.

Treesort. Understand how to use a BST to sort items. Insert all the items, then perform an inorder traversal. This algorithm is similar to quicksort, where each node corresponding to a partitioning item that subdivides its descendants into smaller keys and larger keys.

Deletion and Hibbard deletion. Understand how to delete an item with 1 or 0 children. Understand how to delete an item with 2 children, a.k.a. Hibbard deletion.

Symbol table performance. Understand the performance of a BST based symbol table in the best, worst, and average (shuffled) case, with an emphasis on the fact that the height of a tree is the key factor. Understand what Hibbard deletions do to tree height. You do not need to know why Hibbard deletes result in sqrt(n) height.

Recommended Problems

C level

1. What is the best case BST height? Worst case?
2. If shuffling guarantees log n tree height (probabilistically), why don't we simply shuffle our input data before building our BST based symbol table to avoid worst case behavior?
3. Textbook 3.2.3, but give only two orderings.
4. Textbook 3.2.4
5. From the trees shown in figure A below, circle the correct binary tree (not necessarily a BST) that would produce both of the following traversals:

     In-order: AQVNBRMSP
     Pre-order: BQAVNRSMP

   
   
   From the trees shown in figure B below, circle the correct Binary Search Tree that would produce the following traversal: Post-order: ABCDEFG
   Answers
   In figure A, the first or leftmost tree. In figure B, the first or leftmost tree.

B level

1. Stable 2-way partitioning is a partitioning procedure in which items that are smaller than the pivot are kept in the same relative order after partitioning, and likewise with larger items. For example, if we stably partition GADFLY on G, we'd get ADFGLY. Perform an entire quicksort using stable partitioning on the array [5, 3, 2, 1, 9, 7, 6], and record the compares you make along the way. Build the BST for this array. Observe that the same exact compares were made. What are the advantages and disadvantages of this stable 2-way partitioning algorithm compared to the version we discussed in the Quicksort lecture?
2. Textbook 3.2.22 (pretty easy, but good to do)
3. Textbook 3.2.23
4. Textbook 3.2.24 (so easy it might seem hard)
5. If you know that a tree is a BST, which of the following is or is not always sufficient to reconstruct it? For each one, write yes if it is enough to reconstruct the tree, or no if it is not.

     Pre-order traversal:
     In-order traversal:
     Post-order traversal:
     Level-order traversal:

   
   Answers
   Yes, no, yes, yes. Or put another way, only in-order is not sufficient.

A level

1. Write your own pre-order or post-order traversal code.