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A Binary Search Treeis a binarytree with the following properties:

All items in the left subtree are less than

the root.

All items in the right subtree are greateror equal to the root.

Each subtree is itself a binary search tree.

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Basic Property

In a binary search tree,

the left subtree contains key values lessthan the root

the right subtree contains key valuesgreaterthan or equalto the root.

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7-1 Basic Concepts

Binary search trees provide an excellent structure for searching

a list and at the same time for inserting and deleting data into the

list.

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(a), (b) - complete and balanced trees;

(d)nearly complete and balanced tree;

(c), (e)neither complete nor balanced trees

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7-2 BST Operations

We discuss four basic BST operations: traversal, search, insert,

and delete; and develop algorithms for searches, insertion, and

deletion.

Traversals

Searches

Insertion

Deletion

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Preorder Traversal

23 18 12 20 44 35 52

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Postorder Traversal

12 20 18 35 52 44 23

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Inorder Traversal

12 18 20 23 35 44 52

Inorder traversal of a binary search tree produces a

sequenced list

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Three BST search algorithms:

Find the smallestnode

Find the largestnode

Find a requestednode

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BST Insertion

To insert data all we need to do is followthe branches to an empty subtree andthen insert the new node.

In other words, all inserts take place at aleaf or at a leaflike nodea node that hasonly one null subtree.

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30

30 30

30

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Deletion

There are the following possible cases when we delete anode:

The node to be deleted has no children. In this case, allwe need to do is delete the node.

The node to be deleted has only a right subtree. Wedelete the node and attach the right subtree to thedeleted nodes parent.

The node to be deleted has only a left subtree. Wedelete the node and attach the left subtree to thedeleted nodes parent.

The node to be deleted has two subtrees. It is possibleto delete a node from the middle of a tree, but the resulttends to create very unbalanced trees.

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Deletion from the middle of a tree

Rather than simply delete the node, wetry to maintain the existing structure asmuch as possible by finding data to take

the place of the deleted data. This can bedone in one of two ways.

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Deletion from the middle of a tree

We can find the largest node in thedeleted nodes left subtree and move itsdata to replace the deleted nodes data.

We can find the smallest node on thedeleted nodes right subtree and move itsdata to replace the deleted nodes data.

Either of these moves preserves theintegrity of the binary search tree.

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(continued)

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27 27

27 27

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7-3 Binary Search Tree ADT

We begin this section with a discussion of the BST data structure

and write the header f i le for the ADT.

Data StructureAlgorithms

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Homework:Preparation for the final test

Chapter 6 (pp. 265-282)

Chapter 7 (Sections 7.1; 7.2)

p. 292: ex. 1; p. 293: ex. 6, ex. 12

P. 337: ex. 3, ex. 4, ex.6, ex. 7

P. 338: ex. 13, ex. 14

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