Here are the steps to construct the binary search tree from the given pre-order and in-order traversals: 1. The root node is the first element of pre-order traversal, which is 'a'. 2. Search for 'a' in in-order traversal. Elements before 'a' are in left subtree and elements after are in right subtree. 3. Recursively construct left subtree with pre-order elements 'b,c,d' and in-order elements 'c,d,b'. 4. Recursively construct right subtree with pre-order elements 'e,g,h,j' and in-order elements 'h,g,j,e'.

1. BY
A. RAJITHA
DL in Computer Science
TTWRDC- SpARK, NAGARAM-

2.  What is Non Linear Data Structure?
 What is importance of Non Linear Data Structures
 Different Non Linear Data Structures
 Operations of each Non Linear Data Structures
 Applications of each Non Linear Data Structures
 Questioner

3.  A Non-Linear Data structures is a data structure in which
data item is connected to several other data items.
 Non-Linear data structure may exhibit either a
hierarchical relationship or parent-child relationship.
 In non-linear data structures, every data element may
have more than one predecessor as well as successor.
 Elements do not form any particular linear sequence.

4. Non-linear data structures are capable of expressing
more complex relationships than linear data structures.
The memory is utilized efficiently in the non-linear data
structure where linear data structure tends to waste the
memory.
One big drawback of linked list is, random access
is not allowed.

5.  Trees
A tree is a non-empty set one element of which is
designated the root of the tree while the remaining
elements are partitioned into non-empty sets each of which
is a sub-tree of the root.
 Graph
A graph G is a discrete structure consisting of nodes
(vertices) and the lines joining the nodes (edges).

6. Trees are non linear data structure
that can be represented in a
hierarchical manner.
 A tree contains a finite
non-empty set of elements.
 Any two nodes in the tree are
connected with a relationship
of parent-child.
 Every individual elements in a tree
can have any number of sub trees.
 It represents the nodes
connected by edges

7. Root : The basic node of all nodes in the tree. All
operations on the tree are performed with passing root
node to the functions.
Child : a successor node connected to a node is called
child. A node in binary tree may have at most two children.
 Parent : a node is said to be parent node to all its child
nodes.
Leaf : a node that has no child nodes.
Siblings : Two nodes are siblings if they are children to
the same parent node.

8. Ancestor : a node which is parent of parent node ( B is
ancestor node to D,E and F ).
Descendent : a node which is child of child node ( D, E
and F are descendent nodes of node B )
 Level : The distance of a node from the root node, The
root is at level – 0,( B and C at Level 1 and D, E, F have
Level 2 ( highest level of tree is called height of tree )
 Degree : The number of nodes connected to a particular
parent node.
Path : Path is a number of successive edges from source
node to destination node.

9. Edge : Edge is connection between one node to another node
Path : Path is a number of successive edges from source node
to destination node.
Edge : Edge is connection between one node to another node.
Height of a node : It represents number of edges on the
longest path between that node and a leaf.
Height of a Tree : It represents the height of root node.
Depth of a Tree: Depth of a tree represents the number
edges from the tree’s root node to the node.

10.  Free tree
Rooted tree
 Ordered tree
 Regular tree
 Binary tree
Complete Binary tree
Binary Search tree

11.  A free tree is a connected, acyclic graph .
 It is undirected graph
 It has no node designated as a root
 As it is connected, any node can be reached from any
other node by a unique path
Figure: Free Tree

12.  Unlike free tree, rooted tree is a directed graph in which
one node is designated as root, whose incoming degree is
zero
 And for all other nodes incoming degree is one.
Figure: Rooted Tree

13.  In many applications the relative order of the nodes at any
particular level assumes some significance
 It is easy to impose an order on the nodes at a level by
referring to a particular node as the first node, to another
node as the second, and so on
 Such ordering can be
done left to right
Ordered Tree

14.  A tree in which each branch node vertex has the same
out degree is called as Regular Tree.
 If in a directed tree, the out degree of every node is
less than or equal to m, then the tree is called as an
m-ary tree.
 If the out degree of every node is exactly equal to m
(branch nodes) or zero (leaf nodes) then the tree is
called as regular m-ary tree.

15. In a binary tree, each node can
have at most two children.
A binary tree is either empty or
consists of a node called the root
together with two binary trees
called the left subtree and
the right subtree.
 Node with 2 children are called internal nodes and
nodes with 0 children are called external nodes

16.  Assigning level numbers and Numbering of nodes
for a binary tree:
The nodes of a binary tree can be numbered in a
natural way
level by level
left to right.
 The nodes of a complete binary tree can be numbered so
that the root is assigned the number 1.
 A left child is assigned twice the number assigned its
parent.
 A right child is assigned one more than twice the number
assigned its parent.

18.  If h = height of a binary tree, then
a. Maximum number of leaves = 2h
b. Maximum number of nodes = 2h + 1 - 1
 If a binary tree contains m nodes at level l, it contains at
most 2m nodes at level l + 1.
 Since a binary tree can contain at most one node at
level 0 (the root), it can contain at most 2l node at level l
 The total number of edges in a full binary tree with n node
is n - 1.

19.  If every non-leaf node in a binary
tree has nonempty left and right sub trees, the tree is
termed a strictly binary tree.
 A strictly binary tree with n
leaves always contains 2n – 1
nodes.

20.  A full binary tree of height h has all its leaves at level h.
 All non leaf nodes of a full binary tree have two children,
and the leaf nodes have no children.
A full binary tree with height h has 2h + 1 - 1 nodes. A full
binary tree of height h is a strictly binary tree all of whose
leaves are at level h.
For example, a full
binary tree of height 3
contains 23+1 – 1 = 15 nodes.

21.  A binary tree with n nodes is said to be complete if it
contains all the first n nodes of the above numbering
scheme.
A complete binary tree of height h looks like a full binary
tree down to level h-1, and the level h is filled from left to
right.
Leaf nodes at level n
occupy the leftmost
positions in the tree

22.  A Binary tree is Perfect Binary Tree in which all internal
nodes have two children and all leaves are at same level.
 A Perfect Binary Tree of height h has 2h – 1 non leaf
nodes.

23. Array Representation
In array representation of a binary tree, we use one-
dimensional array (1-D Array) to represent a binary tree.
This numbering can start from 0 to (n-1) or from 1 to n.
Lets derive the positions of nodes and their parent and child
nodes in the array.
When we use 0 index based sequencing,
Suppose parent node is an index p.
Then, the left child node is at index (2*p)+ 1.
The right child node is at index (2*p) + 2.
Root node is at index 0.
left_child is at index 1.
Right_child is at index 2.

24. When we use 1 index based sequencing,
Suppose parent node is at index p,
Right node is at index (2*p).
Left node is at index (2*p)+1.
Root node is at index 1.
Left child is at index 2.
Right child is at index 3.
0 1 2 3 4 5 6 7 8 9 10 11………………..20

25. ADVANTAGES
 Any node can be accesses from any other node by
calculating the index.
 Here, data are stored without any pointers to their successor
or ancestor.
DISADVANTAGES
 Other than full binary trees, majority of the array entries may
be empty
 A new node to it or deleting a node from it are inefficient
with this representation

26. Linked List Representation
 Binary trees can be represented by links where each node
contains the address of the left child and the right child.
If any node has its left or right child empty then it will have in
its respective link field, a null value.
A leaf node has null value in both of its links.
Node Structure in Binary Tree

28. ADVANTAGES
 The drawback of sequential representation are overcome
in this representation .
 We may or may not know the tree depth in advance. Also
for unbalanced tree, memory is not wasted.
 Insertion and deletion operations are more efficient in
this representation.

29. DISADVANTAGES
 In this representation, there is no direct access to any
node .
 It has to be traversed from root to reach to a particular
node
 As compared to sequential representation memory
needed per node is more.
 This is due to two link fields (left child and right child
for binary trees) in node

30. Creation : Creating an empty binary tree to which ‘root‘ points .
Traversal : To Visiting all the nodes in a binary tree
Deletion : To Deleting a node from a non-empty binary tree
Insertion : To Inserting a node into an existing/empty binary tree.
Merge : To Merging two binary trees
Copy : Copying a binary tree .
Compare : Comparing two binary trees.
Find replica or mirror

31.  A binary tree is defined recursively: it consists
of a root, a left sub tree, and a right sub tree.
 To traverse (or walk) the binary tree is to visit each
node in the binary tree exactly once.
 Tree traversals are naturally recursive.
 Standard traversal orderings:
• preorder
• inorder
• postorder

33. In Preorder, the root
is visited before (pre)
the subtrees traversals.
 In Inorder, the root is
visited in-between left
and right subtree traversal.
 In Postorder, the root
is visited after (post)
the subtrees traversals

34. Assume: visiting a node
is printing its data
• Preorder: 15 8 2 6 3 7 11 10
12 14 20 27 22 30
• Inorder: 2 3 6 7 8 10 11 12
14 15 20 22 27 30
• Postorder: 3 7 6 2 10 14 12 11
8 22 30 27 20 15

35.  Stores keys in the nodes in a way so that searching,
insertion and deletion can be done efficiently.
Binary Search Tree Property
For every node X, all the keys in
its left subtree are smaller than
the key value in X, and all the
keys in its right subtree
are larger than the key value in X

36. Binary Tree Non Binary Tree

38.  If we are searching for 15,
then we are done.
If we are searching for a
key < 15, then we should
search in the left subtree.
If we are searching for a
key > 15, then we should
search in the right subtree.

40.  FindMinimum: Start at the root and go left as
long as there is a left child. The stopping point is
the smallest element
FindMaxmum: Start at the root and go right as
long as there is a right child. The stopping point is
the largest element
Ascending order: Inorder traversal of a binary
search tree gives prints elements in sorted order.

41.  Storing naturally hierarchical data
• File System
• Organizational Structure of an institution
• Class inheritance tree
 Organize data for quick search, insertion, deletion-
Binary Search Tree.
 Dictionary
 Networking Routing Algorithm.
Problem representation
• Expression trees
• Decision tree.

42. 1. Construct the tree from the following preorder and inorder
travels: Preorder: a,b,c,d,e,g,h,j,f. Inorder: c,d,b,a,h,g,j,e,f.

43. Construct a binary search tree with the below information.
The preorder traversal of a binary search tree 10, 4, 3, 5, 11, 12.
Preorder Traversal is 10, 4, 3, 5, 11, 12. Inorder Traversal of Binary
search tree is equal to ascending order of the nodes of the Tree.
Inorder Traversal is 3, 4, 5, 10, 11, 12.
The tree constructed using
Preorder and Inorder
traversal is