Introduction and basic of Trees and Binary Trees

Tree & Binary Tree
19CSE212 : Data Structures and Algorithms
Dr. Chandan Kumar
Department of Computer Science and Engineering
Jan - Jun 2024
Dr Chandan Kumar Jan - Jun 2024

Contents
1 Introductionto Tree
3
4
2 Binary Tree & Its Types
OperationsonBinaryTree
5
Tree Traversal (Pre-order, In-order & Post-order)
8
Constructing a Binary Tree from Traversal Results

Tree
• So far, we have discussed mainly linear data
structures – arrays, linked lists, stacks and queues.
• Now we will discuss a non-linear data structure
called tree.
stack

Tree
• A tree is recursively defined as a set of one or more
nodes where one node is designated as the root of the
tree and all the remaining nodes can be partitioned
into non-empty sets each of which is a sub-tree of the
root.
• Trees are mainly used to represent data containing a
hierarchical relationship between elements, for
example, records, family trees and table of contents.

Tree
Tree A Hypothetical Family
Tree

Basic Terminology
• Root node: The root node R is the topmost node in the tree. If R = NULL,
thenitmeansthetreeisempty.
• Sub-trees: If the root node R is not NULL, then the trees T1, T2, and T3 are
called thesub-treesofR.
• Leaf node: Anode that has no children is called the leaf node or the terminal
node
• Path: Asequence of consecutive edges is called a path. For example, in Fig.,
thepathfromtherootnodeAtonodeIisgivenas:A,D,andI.
• Ancestor node: An ancestor of a node is any predecessor node on the path
from root to that node. The root node does not have any ancestors. In the tree
giveninFig.,nodesA,C,andGaretheancestorsofnodeK.

Basic Terminology
• Descendantnode:Adescendant nodeis anysuccessornodeon anypath from
the node to a leaf node. Leaf nodes do not have any descendants. In the tree
giveninFig.,nodesC,G,J,andKare thedescendantsofnodeA.
• Level number: Every node in the tree is assigned a level number in such a
way that the root node is at level 0, children of the root node are at level
number 1. Thus, every node is at one level higher than its parent. So, all child
nodeshavealevel numbergivenbyparent’slevelnumber+1.
• Degree: Degree of a node is equal to the number of children that a node has.
Thedegreeofaleafnodeiszero.

Basic Terminology

Binary Tree
• A binary tree, is a tree in which no node can have more than
two children. The topmost element is called the root node, and
each node has 0, 1, or at the most 2 children.
• Consider a binary tree T,
here ‘A’ is the root node
of the binary tree T.
• ‘B’ is the left child of ‘A’ and
‘C’ is the right
child of ‘A’
• i.e A is a father of B and C.
• The node B and C are called
siblings.
• Nodes D,H,I,F,J are leaf node

Binary Tree
• A binary tree is a finite set of elements that are either empty or
is partitioned into three disjoint subsets.
• The first subset contains a single element called the root of the
tree.
• The other two subsets are themselves binary trees called the
left and right sub-trees of the original tree.
• A left or right sub-tree can be empty.
• Each element of a binary tree is called a node of the tree.

Binary Tree
• The following figure shows a binary tree with 9 nodes where A
is the root.
• The root node of LA is node B, the root node of RA is C and
so on
• Since a binary tree can contain at most 1 node at level 0 (the
root), it contains at most 2L nodes at level L.
• The left sub tree of the root
node, which we denoted by
LA, is the set LA = {B,D,E,G}
and the right sub tree of the
root node, RA is the set
RA={C,F,H,I}

Types of Binary Tree
Types of Binary Tree based on the number of children:
1. Full Binary Tree
2. Degenerate Binary Tree
3. Skewed Binary Trees
Types of Binary Tree on the basis of the completion of levels:
1. Strictly binary tree
2. Complete binary tree
3. Almost complete binary tree

• Full Binary Tree: A Binary Tree is a full
binary tree if every node has 0 or 2 children.
• Degenerate (or pathological) tree: A Tree where
every internal node has one child.
• A skewed binary tree: It is a
pathological/degenerate tree in which
the tree is either dominated by the left
nodes or the right nodes.

• Strictly binary tree: If every non-leaf node in a binary tree has
nonempty left and right sub-trees, then such a tree is called a
strictly binary tree.
• Or, to put it another way, all
of the nodes in a strictly binary
tree are of degree zero or two,
never degree one
• A strictly binary tree with
N leaves always contains
2N – 1 nodes.

• A complete binary tree is a binary tree in which every level,
except possibly the last, is completely filled, and all nodes are as
far left as possible
• A complete binary tree of depth d is called strictly binary tree
if all of whose leaves are at level d.
• A complete binary tree has
2d nodes at every depth d and
2d -1 non leaf nodes.

• An almost complete binary tree is a tree where for a right
child, there is always a left child, but for a left child there may
not be a right child.
Fig. Almost complete binary tree but not a Strictly
binary tree. Since node E has a left son but not a
right son

Operations on Binary Tree

Node of Binary Tree using Java
class Node {
int key;
Node left, right;
public Node(int item)
{
key = item;
left = right = null;
}
}

Tree Traversal
• Traversal is a process of visiting all the nodes of a tree and may
print their values too.
• All nodes are connected via edges (links) we always start from
the root (head) node.
• There are three ways which we use to traverse a tree:
i. Pre-order Traversal
ii. In-order Traversal
iii. Post-order Traversal
• Pre-order: <root><left><right>
• In-order: <left><root><right>
• Post-order: <left><right><root>
• Generally, we traverse a tree to search or locate a given item or
key in the tree or to print all the values it contains.

Pre-order Traversal
• The preorder traversal of a nonempty binary tree is defined as
follows:
 Visit the root node
 Traverse the left sub-tree in preorder
 Traverse the right sub-tree in preorder

In-order Traversal
• The in-order traversal of a nonempty binary tree is defined as
follows:
 Traverse the left sub-tree in in-order
 Traverse the right sub-tree in inorder
The in-order traversal output of the given tree is H D I B E A F C G

Post-order Traversal
• The post-order traversal of a nonempty binary tree is defined as
follows:
 Traverse the left sub-tree in post-order
 Traverse the right sub-tree in post-order
The post-order traversal output of the given tree is H I D E B F G C A

Example
The outcome of different traversals on the above tree:
• Pre-order traversal: 7 5 10 31 11 23 15
• In-order traversal: 10 5 7 11 23 31 15
• Post-order traversal: 10 5 23 11 15 31 7
• Level-order traversal: 7 5 31 10 11 15 23

Example
• Pre-order traversal:
a) A, B, D, G, H, L, E, C, F, I, J, K b) A, B, D, C, E, F, G, H, I
• In-order traversal:
a) G, D, H, L, B, E, A, C, I, F, K, J b) B, D, A, E, H, G, I, F, C
• Post-order traversal:
a) G, L, H, D, E, B, I, K, J, F, C, A b) D, B, H, I, G, F, E, C, and A

Example: Level-order Traversal

• We can construct a binary tree if we are given at least two traversal
results. The first traversal must be the in-order traversal and the
second can be either pre-order or post-order traversal.
• The in-order traversal result will be used to determine the left and
the right child nodes, and the pre-order/post-order can be used to
determine the root node.
Step 1: Use the pre-order sequence to determine the root node of the
tree. The first element would be the root node.
Step 2: In an in-order traversal sequence, elements to the left of the
root node constitute the left subtree, while those to the right form the
right subtree.
Step 3: Recursively select each element from pre-order traversal
sequence and create its left and right sub-trees from the in-order
traversal sequence.

Example:
In–order Traversal: D B E A F C G
Pre–order Traversal: A B D E C F G

Example:
Input: inorder = [9,3,15,20,7], postorder = [9,15,7,20,3]

Example:
Inorder:[ 40, 20 , 50, 10, 60, 30], Postorder: [40, 50, 20, 60, 30, 10]

List of References
https://www.javatpoint.com/
https://www.tutorialspoint.com
https://www.geeksforgeeks.org/
https://www.programiz.com/dsa/
https://chat.openai.com/

Introduction and basic of Trees and Binary Trees

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