Meljun Cortes Data Structures Trees

8/21/2019 Meljun Cortes Data Structures Trees

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Data Structures

Trees

root is on the top and the leaves are at thebottom

displays data items in a hierarchical structure

organize information in database systems

also work as ordered array and a linked list

Trees *Property of STI Page 1 of 34

A tree is composed of nodes connected byedges or lines


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Data Structures

Basic Terminologies

Nodes

Root Node

Edges / lines / paths

Leaf nodes

Internal Nodes / Child Nodes

Subtree

Visiting

Traversin


Levels

Degree

Depth of the tree

Keys


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Data Structures

Exercise

Given the tree below, find the following:

1. Root node

2. Depth of the tree

3. Nodes in level 4

4. Degree of D5. Child/Children of Z

6. Total Nodes in the tree

7. Parent of F


8. Leaf Nodes

9. Child nodes

10. Maximum degree of the tree


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Data Structures

Binary Tree

every node in a tree can have at most twochildren

Left subtree

Right subtree


two applications of binary tree

binary expression tree

binary search tree


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Data Structures

Binary Search Tree

D:\Annual\Martin\Inventory\January.dat.

Hierarchical file structure in a computer systemis one of the common trees

Hierarchical file structure is not a binary tree

Binary search trees are trees such that any

node has a key which is no more than the keyin its right child node and no less than the key


Figure A – BinarySearch Tree

Figure B – NOT a BST


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Data Structures

Operations on Binary Tree

Node Class

class Node

{

int iData; //data used as key value

float fData; //other datanode leftChild; //node’s left child

node rightChild; //node’s right child

public void displayNode()


{

}

{

class Node

{

person p1; //reference to person object

node leftChild; //node’s left child

node rightChild; //node’s right child

}

class Person{

int iData;

float fData;

}


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Data Structures


Tree Class

class Tree

{

private Node root; //the only data field

//in Tree

public void find(int key)

{


}

public void insert(int id, double dd)

{

}

public void delete(int id)

{

}

//you may include other methods

}


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Data Structures


TreeApp Class

class TreeApp

{ public static voids main(String[] args)

{

Tree theTree = new Tree; //make a tree

theTree.insert(50, 1.5); //insert 3 nodes


theTree.insert(25, 1.7);

theTree.insert(75, 1.9);

node found = theTree.find(25); //find

//node with key 25

if(found != null)

System.out.println(″Found the node with

key 25″);

else

System.out.println(″Could not find node

with key 25″);

}

}


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Data Structures


Finding a Node

pubic Node find(int key)

{ Node current = root

while(current.iData != key)

{


.

current = current.leftChild;

else

current = current.rightChild if(current == null)

return null;

}

return current;

}


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Data Structures

Inserting a Node


public void insert(int id, double dd)

{

Node newNode = new Node();

newNode.iData = id;

newNode.dData = dd;

if(root==null)

root = newNode;else

{

Node current = root;

Node parent;

while(true)

{


parent = current;

if(id < current.iData)

{

current = current.leftChild;

if(current == null){

parent.leftChild = newNode;

return;

}

}

else

{

current = current.rightChild;

if(current == null)

{

parent.rightChild = newNode;

return;

}

}

}

}

}


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Data Structures

a binary tree used in representing arithmeticexpression

specifically the combinations of operators,

operands, and order of evaluation

Binary Expression Tree



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Data Structures

Constructing an Expression Tree

Read the symbols one at a time.

If the symbol is an operand:

Create a one-node tree.

Push a pointer to the node into thestack.

If the symbol is an operator:

3.1 Pop two pointers from stack. These



po n ers represen e a ress o eroot nodes of two trees, T1 and T2.

3.2 Form a new tree whose root is theoperator and whose left and rightchildren are T1 and T2, respectively.

3.3 Push the pointer to the new tree intothe stack.

Repeat steps 1 thru 3 until the last symbol

has been read and processed.


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Data Structures

Example: AB+CDE+∗∗∗∗/

“A” pushed into stack

“B” pushed into stack



subtree for “+” created

“+” pushed into stack


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Data Structures

“C” pushed into stack

“D” and “E” pushed into stack



subtree for “+” created


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Data Structures

“+” pushed into stack

subtree for “*” created



“*” pushed into stack


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Data Structures

subtree for “/” created

“/” ushed into stack




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Data Structures

Create a binary expression tree for the infixnotation given below.

1. ( ( A + B ) - ( C / D ) ) + ( E * ( F / D ) )

2. A + ( B * ( C – D ) ) – ( E – F * ( G + H ) )

Exercise



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Data Structures

Binary Tree Traversal

to visit all the nodes of the tree in order

not commonly used as finding, inserting,deleting nodes

useful in some circumstances and simpler thandeleting of nodes

Inorder

Preorder

Postorder

Inorder Traversal


1. e no e as a e su ree:

Traverse the left subtree in inorder

(recursive call). Once completed,proceed to step 2.

Otherwise, proceed to step 2.

2. Read the root node.

3. If the node has a right subtree:

Traverse the right subtree in inorder(recursive call).

The nodes of the tree willbe visited in the order:

A B C D E F G


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Data Structures


Preorder Traversal


2. If the node has a left subtree:

Traverse the left subtree in preorder

(recursive call). Once completed,proceed to step 3.




raverse e r g su ree n preor er(recursive call).

The nodes of the treewill be visited in the

order:

D B A C F E G


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Data Structures


Postorder Traversal

1. If the node has a left subtree:

Traverse the left subtree in postorder(recursive call). Once completed,

proceed to step 2.Otherwise, proceed to step 2.


Traverse the right subtree in postorder


. ,proceed to step 3.



The nodes of the tree

will be visited in the

order:A C B E G F D


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Data Structures

Generating the given expression tree into thethree traversal methods.

Inorder Sequence

Inorder Traversal



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Data Structures

Inorder Traversal



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Data Structures

Inorder Traversal



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Data Structures

Inorder Traversal



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Data Structures

Preorder Traversal



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Data Structures

Preorder Traversal



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Data Structures

Preorder Traversal



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Data Structures

Preorder Traversal



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Data Structures

Postorder Traversal



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Data Structures

Postorder Traversal



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Data Structures

Postorder Traversal



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Data Structures

Postorder Traversal



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Data Structures

Inorder Traversal Code

private void inOrder(node localRoot)

{if(localRoot != null)

{

inOrder(localRoot.leftChild);

localRoot.displayNode();


inOrder(localRoot.rightChild);

}

}


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Data Structures

Exercise

Given the expression tree below, generate thethree traversal methods.

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Meljun Cortes Data Structures Trees