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By Dawn J. LawrieLoyola College in Maryland

Lecture 14

Binary Search Trees

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The ADT Binary Search Tree

A deficiency of the ADT binary tree whichis corrected by the ADT binary search tree Searching for a particular item

Record A group of related items, called fields, that are

not necessarily of the same data type

Field A data element within a record

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A data item in a binary search tree has aspecially designated search key

A search key is the part of a record thatidentifies it within a collection of records

KeyedItem class

Contains the search key as a data field and a

function for accessing the search key

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Figure 10.18 UML diagram for the class BinarySearchTree

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Algorithms for the ADT Binary

Search Tree Operations A recursive search function

Searches the binary search tree bintree forthe item whose search key is searchKey

Searches recursively in the right or left subtree(depending on the value of the item) untilsearchKeymatches the search key of the nodesitem

search(in binTree:BinarySearchTree,in searchKey:KeyType);

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ADT Binary Search Tree: Insertion

Inserts newItem into the binary searchtree to which treePtr points insertItem(inout

treePtr:TreeNodePtr,in newItem:TreeItemType)

To copy a tree Traverse it in preorder and insert each item

visited into a new tree

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ADT Binary Search Tree: Insertion

Figure 10.23

(a) Insertion into an

empty tree;

(b) search terminates at

a leaf;

(c) insertion at a leaf

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Exercise

Beginning with an empty binary searchtree, what binary search tree is formedwhen you insert the following values in the

order given: J, N, B, A, W, E, T?

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ADT Binary Search Tree: Deletion

Three possible cases for deleting node N

Nis a leaf Set the pointer in Ns parent to NULL

Nhas only one child

Let Ns parent adopt Ns child

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ADT Binary Search Tree: Deletion

Three possible cases for deleting node N(continued)

Nhas two children

Locate another node Mthat is the leftmost node inNsright subtree

Mssearch keyis called the inorder successor of Nssearch key

Copy the item that is in Mto N Remove the node Mfrom the tree

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ADT Binary Search Tree: Retrieval and

Traversal

The retrieval operation can beimplemented by refining the search

algorithm

Return the item with the desired search key ifit exists

Otherwise, throw TreeException

Traversals for a binary search tree are thesame as the traversals for a binary tree

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ADT Binary Search Tree

Theorem 10-1The inorder traversal of a binary searchtree Twill visit its nodes in sorted search-

key order Theorem 10-2

A full binary tree of height h 0 has 2h 1

nodes Theorem 10-3

The maximum number of nodes that abinary tree of height h can have is 2h 1

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The Efficiency of Binary Search

Tree Operations

Theorem 10-4

The minimum height of a binary tree with n

nodes is log2(n+1) The height of a particular binary search

tree depends on the order in which

insertion and deletion operations areperformed

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The Efficiency of Binary Search

Tree Operations

Figure 10.32 Counting the nodes in a full binary tree of height h

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Applications

Algorithms for saving a binary search tree Saving a binary search tree and then

restoring it to its original shape

Uses preorder traversal to save the tree to a file Saving a binary tree and then restoring it to a

balanced shape Uses inorder traversal to save the tree to a file

Can be used if the data is sorted and the numberof nodes in the tree is known

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The STL Search Algorithms for

Sorted Ranges

binary_search Returns true if a specified value appears in

the sorted range

lower_bound; upper_bound Returns an iterator to the first occurrence;

or to one past the last occurrence of a

value equal_range

Returns a pair of iterators that indicate thefirst and one past the last occurrence of avalue

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General Trees

An n-ary tree

A generalization of a binary tree whosenodes each can have no more than n

children

Figure 10.38A general tree

Figure 10. 41

An implementation of the n-ary tree in Figure 10.38

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Summary

Binary trees provide a hierarchicalorganization of data

The implementation of a binary tree isusually pointer-based

A client-defined visit function to thetraversal operation can customize theoperations on the items of the tree

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Summary

The binary search tree allows you to use abinary search-like algorithm to search foran item with a specified value

Binary search trees come in many shapes The height of a binary search tree with n

nodes can range from a minimum of log2(n+1) to a maximum of n

The shape of a binary search tree determinesthe efficiency of its operations

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Summary

An inorder traversal of a binary search treevisits the trees nodes in sorted search-keyorder

The treesortalgorithm efficiently sorts anarray by using the binary search trees

insertion and traversal operations

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Summary

Saving a binary search tree to a file

To restore the tree as a binary search tree ofminimum height

Perform inorder traversal while saving the tree to afile

To restore the tree to its original form

Perform preorder traversal while saving the tree toa file

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Final Review

Exam is cumulative

This will be a 2 hoursexam

Topics Binary Search Trees

Trees

Big-O notation

Sorting

Inheritance

More Topics

Linear Data Structures

Manipulating Linked

Lists Allocating/Deallocating

memory

Memory Layout and

the runtime stack Exceptions

Recursion

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