Binary Search Trees

1
Topic 13

• Binary Search Trees

2
Chapter Objectives

• define a binary search tree abstract data
  structure
• demonstrate how a binary search tree can be used
  to solve problems
• examine various binary search tree
  implementations
• compare binary search tree implementations

3
Binary Search Tree

• a binary search tree (BST) is a binary tree with
  an ordering property of its elements
• for each node, its data element is greater than
  that of its left child and less than or equal to
  that of its right child
• example

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Binary Search Tree (contd)

• so, the data in any node is
• greater than the data in any node in its left
  subtree
• less than or equal to the data in any node in its
  right subtree
• examples are these BSTs?

5
Binary Search Tree Operations

• why is it called a binary search tree?
• algorithm to search for an item in the tree
• compare data item to the root of the (sub)tree
• if data item data at root, found
• if data item lt data at root, go to the left if
  there is no left child, data item is not in tree
• if data item gt data at root, go to the right if
  there is no right child, data item is not in tree

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Search Operation

• examples

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Add Operation

• adding an element to a BST requires that the BST
  property be maintained!
• algorithm
• follow the algorithm for searching, until there
  is no child
• insert at that point
• examples

8
Adding Elements to a Binary Search Tree
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Remove Operation

• what happens when you remove an element from a
  BST?

10
Removing elements from a binary search tree
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Removing the minimum element from a binary search
tree
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Binary Search Tree Traversals

• consider the traversals of a binary search tree
• preorder
• inorder
• postorder
• level-order
• is there anything special about the order of the
  data in the BST, for each traversal?
• examples

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Operations on a Binary Search Tree
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UML description of the BinarySearchTreeADT
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Implementing Binary Search Trees

• a BST is just a binary tree with the ordering
  property imposed on all nodes in the tree
• so, we can extend the BinaryTree class

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BST Implementation

• addElement
• removeElement
• removeAllOccurrences
• removeMin
• the rest are left as exercises

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Using Binary Search Trees Implementing Ordered
Lists

• analysis of the BinarySearchTreeOrderedList
  implementation

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The common operations on a list
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The operation particular to an ordered list
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Analysis of linked list and binary search tree
implementations of an ordered list
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A degenerate binary tree