binary search trees

( Fri Oct 2, 2015 )

Binary Search Trees

binary trees, tables, and binary search trees.
- ADT binary tree -- the ADT is determined by these properties
  - Organization -- A binary tree is empty, or consists of a root with a left subtree and a right subtree. The subtrees are binary trees.
  - Elements -- any given homogeneous set
  - Operations -- see the list and specs on pages 439-440 of the sixth edition of Carrano/Henry.
- ADT Dictionary -- see the specs in the sixth edition of Carrano/Henry, Chapter 18.
  - The dictionary is important -- a basic data type for representing many kinds of databases.
  - We can use a simple array-based list or linked list to implement a dictionary, but neither of these implementations will be very efficient if the number of items to be stored is large. (Discuss)
  - The dictionary ADT can be implemented quite efficiently as a binary search tree. This explains the importance of binary search trees.
  - Organization of the dictionary data type -- There is a totally ordered set of keys. Each element of the dictionary contains a data field. Each element is also assigned a unique key. Elements and the data in them are accessed by key value.
  - Elements of a dictionary data type -- data fields from any homogeneous set -- typically each element of the dictionary is a certain kind of record which may contain many different information fields -- an employee record for example.
  - Major Dictionary data type Operations
    - Insert by key value
    - Delete by key value
    - Retrieve by key value
    - Traverse Dictionary
- What is the reason for using BST's? -- they provide a hybrid of the array and linked list versions of a dictionary in which all major operations can be implemented efficiently.
- ADT binary search tree
  - Organization -- A binary search tree is a binary tree in which each node contains a key that is an element of a totally ordered set. For every node N in the tree, all keys in the left subtree of N are less than the key in N, and all keys in the right subtree of N are greater than the key in N.
  - Elements -- any given homogeneous set
  - Operations -- see the discussion on page 444 of the sixth edition of Carrano/Henry for the list and the specs. (They are almost the same as those for the binary tree.)
    - Discuss implementation of search.
```
search (tree: BinarySearchTree; targ: ItemType)
        if (tree is empty) return "not found"
   else if (targ == data item in tree root) return "found"
   else if (targ < data item in tree root)
           search(left subtree of tree, targ)
   else    search(right subtree of tree, targ)
```
    - Discuss implementation of insertion:
      
      Do a search, and insert the node where the search "falls out of the tree"
    - Discuss implementation of traversal:
      
      Traversals work the same way they do for an ordinary binary tree. InOrder traversal visits the nodes in ascending key order.
    - Discuss implementation of deletion.
      - -- when the delete node is a leaf, "find and chop"
      - -- when the delete node has one child, "grandparent adopts"
      - -- when the delete node has two children, "promote the inorder successor"