date structure have a great application .Trees are basic important part of this structure . all basic discreption is provided in this presentation . data structure and trees theory . types of binary search,representation of trees , implementation of trees , important algorithm , extended binary tree , linked list representation of trees. orders in binary trees .

1. 3 TREE
A tree is a (possibly non-linear) data structure made up of nodes or vertices and edges without
having any cycle. The tree with no nodes is called the null orempty tree. A tree that is not
empty consists of a root node and potentially many levels of additional nodes that form a
hierarchy.
Not a tree: cy cle B→C→E→D→B. B has more than one parent (inbound edge
Not a tree: two non-connected parts, A→B and C→D→E. There is more than one root
Not a tree: undirected cy cle 1-2-4-3. 4 has more than one parent (inbound edge
Not a tree: cy cle A→A. A is the root but it also has a parent.
Each linear list is triv ially a tree
# TERMINOLOGIES OF TREES:
1.ROOTS: THE FIRST NODE OF TREE IS CALLED ROOT NODE.

2. 2.PARENTS: In simple words,the node whichhasbranchfromit to anyothernode is calledas
parentnode.Parentnode can alsobe definedas"The node whichhaschild/children".
3. EDGES: the node whichhasa linkfromitsparentnode iscalledaschildnode.Ina tree,any
parentnode can have any numberof childnodes.Ina tree,all the nodesexceptrootare child nodes.
4.CHILD : In simple words,the node whichhasalinkfromitsparentnode iscalledas childnode.
In a tree,anyparentnode can have anynumberof childnodes.
5.SIBLING: a group of nodes with same parents.
6.LEAF: a node with no child.
7. INTERNAL NODE: In a tree data structure,nodesotherthanleaf nodesare called
as Internal Nodes. The rootnode isalsosaidto be Internal Node if the tree hasmore than one
node. Internal nodesare alsocalledas'Non-Terminal'nodes.
8.HEIGHT:(for height of of tree we need to move from downward (root node ) to upward
upto which height of node is asked.) ina tree datastructure,the total numberof egdes fromleaf
node to a particularnode inthe longestpathiscalledas HEIGHT of that Node. Ina tree,heightof the
root node issaidto be heightof the tree.

3. 9. DEGREE: the total numberof childrenof a node iscalledas DEGREE of that Node
10.DEPTH: In a tree data structure,the total numberof egdesfromrootnode to a particular
node iscalledas DEPTH of that Node. Ina tree,the total numberof edgesfromrootnode to a leaf node
inthe longestpathissaidto be Depthof the tree.
11. PATH: the sequence of NodesandEdgesfromone node toanothernode iscalled
as PATH betweenthattwoNodes.
12. SUBTREE: each childfroma node formsa subtree recursively.Everychildnodewillforma
subtree onitsparentnode.
13. LEVEL: ina tree each stepfromtop to bottomiscalledas a Level andthe Level countstarts
with'0' and incrementedbyone ateach level (Step).
Figure 1.
14.DESCENDANT: youjust needtomove formroot node to leaf node (takingrootnode).

4. Consider fig. 1 letwe needtofind descendantand ancestorfor ‘J’,’G’,’k’.
Thenanswerwill be
>>> descendentof ‘j ‘: A,B,E,J & FOR ‘G’ : A,C,G & FOR ‘K’: A,C,G,K.
15.ANCESTOR: youjustneedtomove formgivennode toroot nodes takingall nodes coming
inbetween.(takinggivennode node).(eg.Same problem )
>>ancestor of ‘j ‘: J,E,B,A & FOR ‘G’ : G,C,A & FOR ‘K’: K,G,C,A.
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APPLICATIONS OF TREE :
1. Organizationstructureof corporation.
2. Table of contentof books.
3. Dos or window file system.
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# BINARY TREE
In a normal tree,everynode canhave any numberof children.Binarytree isaspecial type of tree data
structure inwhicheverynode can have a maximumof 2 children.One isknownasleftchildandthe
otheris knownasright child.
TYPE OF BINARY TREES :
1. STRICTLY BINARY TREE
2.COMPLETEBINARYTREE
3. EXTRENDED BINARYTREE
>> STRICTLY BINARYTREE : In a binarytree,everynode canhave a maximumof two children.
But instrictlybinarytree,everynode shouldhave exactlytwochildrenornone.Thatmeansevery
internal node musthave exactlytwochildren.

5. >>2. COMPLETEBINARY TREE(PerfectBinaryTree)
In a binarytree,everynode canhave a maximumof twochildren.Butinstrictlybinarytree,everynode
shouldhave exactlytwochildrenornone andincomplete binarytree all the nodesmusthave exactly
twochildrenandat everylevel of complete binarytree theremustbe 2^n level numberof nodes.For
example atlevel 2there mustbe 2^2 = 4 nodesandat level 3there mustbe 2^3 = 8 nodes.
A binary tree in which every internal node has exactly two children and all leaf
nodes are at samelevel is called Complete Binary Tree.
Complete binarytree isalsocalledas PerfectBinaryTree.
3.EXTRENDED BINARYTREE
A binarytree can be convertedintoFull Binarytree byaddingdummynodestoexistingnodeswherever
required .The full binary tree obtained by adding dummy nodes to a binary tree is called as
Extended Binary Tree.

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# REPRESENTATION OF BINARY TREE:
(normal trees cann’t be represent as array and link list,justbecauseof there irregular shape.)
1. linked list representation
2. Array representation (Let we have this binary tree.)
1. linked list representation
2. Array representation
(1-D) HERE THE ARRANGEMENT OF DATA IS DONE ON THE BASESE OF “ 2*K , 2*K
+1 “ OFCHILDS OF EVERY NODE . ( K IS THE NODE NUMBER GIVEN ,(STARTS FORM 0).

7. ###############################################
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8. ###############################################
# BINARY TREE TRAVERSALS
(TO DISPLAY A BINARY TREE WE NEED TO FOLLOW SOME
ORDERS )
1. INORDER
2. PREORDER
3. POSTORDER
(HOPELLY U ALL KNOW HOW TO DO THESE JUST VERIFY ANSWERS )
1. 1. In - Order Traversal( leftChild - root- rightChild )
I - D - J - B - F - A - G - K - C – H

9. 2. Pre - OrderTraversal ( root - leftChild - rightChild)
A - B - D - I - J - F - C - G - K - H
3. Post - OrderTraversal ( leftChild - rightChild - root)
I - J - D - F - B - K - G - H - C – A
Program for all three method is :
#include <stdio.h>
#include <stdlib.h>
struct node
{ intvalue;
struct node*left;
struct node*right;
};
struct node*root;
struct node*insert(structnode*r,intdata);
voidinOrder(structnode*r);
voidpreOrder(structnode*r);
voidpostOrder(structnode*r);
intmain()
{ root= NULL;
int n,i,v;
printf("Howmanydata'sdoyou wantto insert?n");
scanf("%d",&n);
for( i=0; i<n;i++){
printf("Data%d:",i+1); scanf("%d",&v);

10. root= insert(root,v);}
printf("InorderTraversal:");
inOrder(root);
printf("n");
printf("PreorderTraversal:");
preOrder(root);
printf("n");
printf("PostorderTraversal:");
postOrder(root);
printf("n");
return0; }
struct node*insert(structnode*r,intdata)
{ if(r==NULL)
{ r = (structnode*) malloc(sizeof(structnode));
r->value =data;
r->left=NULL;
r->right= NULL;
}
else if(data<r->value){
r->left=insert(r->left,data);}
else {
r->right= insert(r->right,data);
}returnr; }
voidinOrder(structnode*r)

11. { if(r!=NULL){
inOrder(r->left);
printf("%d",r->value);
inOrder(r->right);
}
}
voidpreOrder(structnode*r)
{ if(r!=NULL){
printf("%d",r->value);
preOrder(r->left);
preOrder(r->right);
}
}
voidpostOrder(structnode*r)
{ if(r!=NULL){
postOrder(r->left);
postOrder(r->right);
printf("%d",r->value); }
}
# Threaded Binary Tree

12. A binarytree isrepresentedusingarrayrepresentationorlinkedlistrepresentation.Whenabinarytree
isrepresentedusinglinkedlistrepresentation,if anynode isnothavinga childwe use NULL pointerin
that position.Inanybinarytree linkedlistrepresentation,there are more numberof NULL pointerthan
actual pointers.Generally,inanybinarytree linkedlistrepresentation,if there are 2N numberof
reference fields,then N+1numberof reference fieldsare filledwithNULL( N+1 are NULL outof 2N ).
ThisNULL pointerdoesnotplayanyrole exceptindicatingthereisnolink(nochild).
Threaded Binary Tree is also a binary tree in which all left child pointers
that are NULL (in Linked list representation) points to its in-order
predecessor, and all right child pointers that are NULL (in Linked list
representation) points to its in-order successor.
(STEPS TO SOLVE THREDED BINARY TREE . (REPRESENT THIS))
1. WRITE INORDER OF TREE.
2. ONE WAY  RIGHT NODE: jointhe inoder successerof of node represented byarrow .
3. TWO WAY -> RIGHT +LEFT NODE : jointhe inorder predecessor.
4. else if nither predecessorof successerispresentof the node thenthatnode isrepresentbyNULL.

13. # HUFFMAN ALGORITHM (IMPORTANT EVEN)
HOPLLY U CAN DO IT
(THIS ALGO IS GREAT , FOR STORE THE DATA IN ANOTHER WAY (BY ITS WAY) EXCEPT OF OUR
OWN )
*HEIGHT OF TREE FORM IS -> N-1 (WHERE THE ‘N’ IS TOTAL NO. OF ELEMENTS TO BE
ARRANGE.)
An example is
Q. arrange these element 2,3,5,7,9,11 using huffman algorithm ?
Steps are:
1. Add smallest elements at level first.
2. Compare answer with given element if
a. If answer is smaller or equal to any element then proceed to add at upper level.
b. If answer is greater then the other elements , proceed to add any other two smaller
numbers(can include answer or another element).
Made by Gaurav Sharma…