Binary Tree Applications - Lecture 32 Section 12people.hsc.edu/faculty-staff/robbk/Coms262/Lectures...Expression Trees Binary Search Trees Searching a BST Inserting into a BST Deleting

Binary TreeApplications

Robb T.Koether

ExpressionTrees

Binary SearchTreesSearching a BST

Inserting into a BST

Deleting from a BST

Count-Balancing aBST

Assignment

Binary Tree ApplicationsLecture 32

Section 12.4

Robb T. Koether

Hampden-Sydney College

Wed, Apr 8, 2009


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Outline

1 Expression Trees

2 Binary Search TreesSearching a BSTInserting into a BSTDeleting from a BSTCount-Balancing a BST

3 Assignment


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Expression Trees

Definition (Expression Tree)A binary expression tree is a binary tree that is used torepresent an expressions with binary operators. Eachinterior node represents an operator. At each interior node,the left subtree represents the left operand and the rightsubtree represents the right operand.

As a consequence, each terminal node represents anoperand.The order of operation is indicated by the structure ofthe tree.


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Expression Trees

For example, (3 + 4) ∗ 5 may be represented as

+

3 4


Robb T.Koether

ExpressionTrees



Deleting from a BST


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Expression Trees

If there is more than one operator, the order ofoperation is indicated by the structure of the tree.

*

+ 5

+

3 *

3 4 4 5


Robb T.Koether

ExpressionTrees



Deleting from a BST


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Traversals and Expression Trees

Perform an in-order traversal of the expression tree andprint the nodes.

*

+ -

4 5 6 3


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Traversals and Expression Trees

Perform a post-order traversal to evaluate theexpression tree.

*

+ -

4 5 6 3


Robb T.Koether

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Deleting from a BST


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Binary Search Trees

Definition (Binary Search Tree)A binary search tree is a binary tree with the followingproperties.

There is a total order relation on the members in thetree.At every node, every member of the left subtree is lessthan or equal to the node value.At every node, every member of the right subtree isgreater than or equal to the node value.


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

A Binary Search Tree

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BinarySearchTree Implementation

The BinarySearchTree class is implemented as asubclass of the BinaryTree class.


Robb T.Koether

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Assignment

Binary Search Tree Interface

Mutatorsvoid insert(const T& value);Insert a new node containing the value into the binarysearch tree.void remove(const T& value);Remove the node containing the value from the binarysearch tree.


Robb T.Koether

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Deleting from a BST


Assignment

Binary Search Tree Interface

Other Member FunctionsT* search(const T& value) const;Search the binary search tree for the value. Return apointer to the node where the value is found. ReturnNULL if the value is not found.void countBalance();Count-balance the binary search tree.


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Searching a BinarySearchTree

Searching a Binary Search Tree

Beginning at the root node, apply the following stepsrecursively.

Compare the value to the node data.If it is equal, you are done.If it is less, search the left subtree.If it is greater, search the right subtree.If the subtree is empty, the value is not in the tree.


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Inserting a Value into a BinarySearchTree

Inserting into a Binary Search Tree

Beginning at the root node, apply the following stepsrecursively.

Compare the value to the node data.If it is less (or equal), continue recursively with the leftsubtree.If is is greater, continue recursively with the rightsubtree.When the subtree is empty, attach the node as asubtree.


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Deleting a Value from a BinarySearchTree

Perform a search to locate the value.This node will have

Two children, orOne child, orNo child.


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Case 1: No ChildDelete the node.


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Delete a Node with No Child

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Delete a Node with No Child

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Case 2: One ChildReplace the node with the subtree of which the child isthe root.


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Delete a Node with Two Children

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Case 3: Two ChildrenLocate the next smaller value in the tree. This value isthe rightmost value of the left subtree.

Move left one step.Move right as far as possible.

Swap this value with the value to be deleted.The node to be deleted now has at most one child.


Robb T.Koether

ExpressionTrees



Deleting from a BST


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Delete a Node with Three Children

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Count-Balancing a BinarySearchTree

Write a function moveNodeRight() that will move thelargest value of the left subtree to the right subtree.

The moveNodeRight() FunctionLocate the largest value in the left subtree.Delete it (but save the value).Place it at the root.Insert the old root value into the right subtree.


Robb T.Koether

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Count-Balancing a BinarySearchTree

Count-Balancing a Tree

Write a similar function moveNodeLeft().Apply either moveNodeRight() or moveNodeLeft()repeatedly at the root node until the tree is balanced atthe root.Then apply these functions recursively, down to theleaves.


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Building a Binary Search Tree

Suppose we wish to transmit the nodes of a balancedbinary search tree to another computer and reconstructthe tree there.In what order should the values be transmitted?


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Building a Binary Search Tree

We could use an in-order traversal to transmit them.At the receiving end, simply call insert() to inserteach value into the tree.The constructed tree will be identical to the original.What do we get if we transmit the values using apre-order traversal?Using a post-order traversal?


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

BinarySearchTree Implementation

BinarySearchTree ClassDownload binarysearchtree.h

Download and run BinarySearchTreeTest.cpp


Robb T.Koether

ExpressionTrees



Deleting from a BST


Assignment

Assignment

HomeworkRead Section 12.4, pages 667 - 694.

Documents

Binary Tree Applications - Lecture 32 Section 12people.hsc.edu/faculty-staff/robbk/Coms262/Lectures...Expression Trees Binary Search Trees Searching a BST Inserting into a BST Deleting