Chapter 7 - Binary Search Tree in the context of DSA.pdf

Bachelor of Information Technology
Chapter(7)
Tree
,

./ Tree
n Concept and Definitions,Basic Operations inBinary Tree,
Tree Height, Level and Depth
OBinary Search Tree, Insertion, Deletion, Traversals, Search
inBST
2

o A tree is a non-linear data structure in which items are
arranged in a sorted sequence.
o It is used to represent hierarchical relationship existing
amongst several data items.
o Recursive definition: A tree is defined as a finite set of
one or more data items (nodes) such that:
There is a special data item called the root ofthe tree.
And its remaining data items are partitioned into a number of
1nutually exclusive (i.e. disjoint) subsets, each ofwhich itself
is a tree (called subtrees)
o Tree data structure grows downwards from top to bottom.
3

Root
i
Key--- p
a
A
,,
, '
, '
'
' t
,
,
,
'
'
'
'
,. '
Siblings
B
,, E G '
,
'
,+•--subtree
� - -- -- .._, - - . - - - - - - - _,
Children
Edge
I
R
C D Height
of the
tree
M
H
Leaf nodes
4

Root
Level 0
B C Level 1
Parent Node E Siblings .. G Level 2
Child Node Level3
Sub-tree Leaf Node
5

1. Each data item in a tree is called a node
2. It is the first data item (or node) in the hierarchical arrangement of data
iten1s. In the above tree, A is the rootiten1
3. The degree ofa node is the number of children of that node
4. The degree of a tree is the maximum degree ofnodes in a given tree
s. Path is the sequence of consecutive edges from source node to
destination node
6. The height of a node is the max path length form that node to a leaf
node
1. The height of a tree is the height ofthe root
8. Depth of a tree is the max level ofany leaf in the tree
9. Level of a node represents the generation of a node. If root node is at
level 0, then its next child node is at level 1, its grandchild is at level 2
and so on.
6

Level 0
B Level 1
Lev-el 2
0 ----------
__.. A is the root node
__.. B is the parent ofE and F
__.. D is the sibling ofB and C
__.. E and F are children ofB
__.. E, F, G, D are external nodes or leaves
__.. A, B, C are internal nodes
__.. Depth ofF is 2
__.. the height of tree is 2
__.. the degree ofnode A is 3
__.. The degree of tree is 3
7

o A binary tree is a finite set of nodes that is either empty or consists of a
root and two disjoint binary trees called the left subtree and the right
subtree.
o Any tree can be transfonned into binary tree.
LI by left child-right sibling representation
o The left subtree and the right subtree are distinguished.
- -
- -
(a) Binary tree
with one node
- -
- - - -
- -
A
- - - -
- - - -
(b) Binary tree (c) Binary tree (d) Binary tree
with two nodes with two nodes with three nodes
8

o Ifevery non-leaf node in a binary tree has nonempty left and right sub
trees, then such a tree is called a strictly binary tree.
o Or, to put it another way, all ofthe nodes in a strictly binary tree are of
degree zero or two, never degree one.
o A strictly binary treewith N leaves always contains 2N - 1 nodes.
B
0 E C
A Strictly Bln11ry Tree
9

o 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.
o A complete binary tree ofdepth d is called strictly bina1y tree ifall ofwhose leaves
are at level d.
D A complete binary tree has 211 nodes at every depth d and 2t1-1 non leafnodes
A complete Binary TrM of depth 3
10

o An almost con1plete binary tree is a tree where for:
.. a right child, there is always a left child,
.J but for a left child there may not be a right child.
8 C
H J
Fig Almost completebinary treebut not strictly binary tree.
Fig Almost complete binarytree. Since node Ehas a left son but not a right soo.
11

o A binary search tree is a binary tree that is either empty or in
which each node possesses a key that satisfies the following three
conditions:
[J For every node X in the tree, the values of all the keys in its left
subtree are smaller than the key value in X.
rJ Keys in its right subtree are greater than the key value in X.
U The left and right subtrees of the root are again binary search trees.
12

Create the binary search tree using the following data elements.
43, 10,79,90, 12,54, 11,9,50
Step 1
Step S
Step 8
Step 2
_ft
®
Step 3
r.-J��,
� �
Step 6
43
Step 4
-------�-
79
'
00 e
St.ep 9
Binary search Tree Creatiol'I
s-i-ep 7
43
fe
05
13

INSERTION
o Inserting a node into an empty tree: In this case, the node inserted into the tree
is considered as the root node.
o Inserting a node into a non-empty tree: In this case, we compare the new node
to the root node of the tree.
.J Ifthe value oftbe new node is less than the value ofthe root node, then if the left
subtree is etnpty, the new node is appended as the left leaf of the root node
else we search continuous down the left subtree.
,.... Tf the value of the new node is greater than the value of the root node, then
if the right subtree is empty, the new node is appended as the right leaf of
the root node else we search continuous do,vn the right subtree.
If the value of the new node is equal to the value of the root node, then
print "DUPLICATE ENTRY" and return.
14

lnsert (TREE, lTEM)
1. IF TREE= NULL
Allocatemen101y for TREE
SET TREE ->DATA=JTEM
SET TRl:E -> LEFT TREE -> RIGHT -NULL
ELSE
IF ITE:1 <TREE-> DATA
lnsert(TREE-> LEFT, ITEM)
ELSE
lnscrt(TREE-> RlGHT. fTEM)
fEND OFIFl
[END OFTF]
2. END
15

ITEM>ROOT·> DATA ROOT - 50
-'"=·="' � 0
Item= 95
tT£M>ROOT·>DATA
ROOT =ROOT ·> "'IQHT
I .
. .. I
0 e� e
�� v
ITl;M>·�OOT -> DATA
ROOT •ROOT·>RIG.HT
STEPl
i
@e
STcEP3
INSERT IT�MTO THERIGHT
OF6S
50
STEP2
STEP4
16

o 13, 3, 4, 12, 14, 10, 5, 1, 8, 2, 7, 9, 11, 6, 18 in that order,
starting from an empty tree insert according to BST
17

SEARCH
./ Search operation is performed with O(log n) time complexity iJ1 a binary search tree.
./ This operation starts from the root node. Jt is used whenever an element is to be searched.
./ The following algorithm shows the search operation in bina1y search tree:
Read the element from the user .
Compare this element with the value of root node in a tree.
Ifelement and value are matching, display "Node is Found" and Lerminate Lbe function.
If element and value are not matching, check whether an element is smaller or larger than a
node value.
lf an clement is smaller, continue the search operation in left subtree.
Ifan element is larger, continue the search operation in right subtree.
Repeal U1c same process until we found the exact clement.
Ifan element wilh search value is found, display "Element is found" and terminate the
function.
I Ifwe reach to a leafnode and the search value is not match to a leafnode, display "Element is
ool found" and terminate the function
18

.; The element to be searched is 10
., Compare the element with the root node l2, IO < 12, hence you move to the left subtree. No need to analyze
the right-subtree
.; Now compare 10 with node 7, 10 > 7, so move to the right-subtree
.; Then compare 10 with the next node, which is 9, 10 > 9, look iu the right subtree child
.; IO matches with the value in the node, IO = I0, return the value to t]le user
..... Elements co be searched
� in the tree .l.O
•10 ·
... I :;c-.,
n1ovP. rl-, rhe
nght: sub-
tit:,P.
.LO <- 1.2 so No need to
.10 > 9 so
move to tho
,1ght S;1..ib tree
ch,lc.1
On comparison
10 matches,
ruturn LhO vuluu
0
19

�&i§�t eg��tY�©� �� [8)�7r
DELETION
1. Locate the node to be deleted and its parent (The
parent of root node is NULL).
2. If leaf node - set NULL to its parent pointer.If
node with one child- redirect the parent pointer
to point to its child pointer.
3. If node with two child- replace node being
deleted by I. Leftmost node of its right subtree II.
OR rightmost node of its left subtree
20

Case 1- Node with zero children: this is the easiest situation, you just need to delete
the node which has 110 fwiher children 011 the right or left.
N00ctocedeleted has0
:hdreo
-
S1mpe)elererre·.oje
ardrerrovetheI•k
1 1. This is the first case ofdeletion
in which you delete a node that
has no children. As you can see
in the diagram that 19, 10 and 5
have no children. But we will
delete 19.
2. Delete the value 19 and remove
the link from the node.
3. View the nevv structure of the
BST vvithout 19
21

Case 2 - Node with one child: once you delete the node, simply connect its child node
with the parent node ofthe deleted value.
A'J:detobedeleleliasl
W010
..
Smoel)rnetierode
anc rep.:er.1·,,ll'me
:M: ,oc,
1. This is the second case ofdeletion in
which you delete a node that has I
child, as you can see in the diagra1n
that 9 has one child.
2. Delete the node 9 and replace it with
its child IO and add a link from 7 to
10
3. View the new structure of the BST
without 9
22

Case 3 Node with two children: this is the most difficult situation, and it works on the
following two rules
• 3a -In Order Predecessor: you need to delete the node with two children and
replace it with the largest value on the left-subtree of the deleted node
I,i:<t,,�"""'0b1ge,i
e-n<t,,tts,b,,,,'llit
IX,j'l)be�II) llll
0
SmpeDl�letie,M?
:1andr;p;,:e�11,;h10
t. Here you will be deleting the
node 12 that has two children
2. The deletion of the node will
occur based upon the in order
predecessor rule, which means
that the largest elen1ent on the
left subtree of 12 will replace it.
3. Delete the node 12 and replace
it with 10 as it is the largest
value on the left subtree
4. View the new structure of the
BST after deleting 12
23

3b - In Order Successor: you need to delete the node with two children and replace it
with the largest value on the right-subtree of the deleted node
hC!Ce·�s��s:.S:r-em
ri1t11'Q'U,b-trc�ct·"'('n.�1c:o
te�ca�
0
S:npleD�etemeIY.}:le
12ind •eplm,rt11,ch:,
f) Re,Jlt
1. Delete a node 12 that has two
children
2. The deletion of the node will
occur based upon the In Order
Successor rule, which means
that the largest element on the
right subtree of 12 will replace
it
3. Delete the node 12 and replace
it with 19 as it is the largest
value on the right subtree
4. View the new structure of the
BST after deleting 12
24

../ Binary tree traversing is a process of accessing every node of
the tree and exactly once. A tree is defined in a recursive
manner. Binary tree traversal also defined recursively.
../ All nodes are connected via edges (links) we always start from
tl1e root(head) node.
../ There are three ways which we use to traverse a tree
,. In-order Traversal
Pre-order Traversal
Post-order Traversal
../ Generally we traverse a h·ee to search or locate given item or
key in the tree or to print all the values it contains.
25

,,, The preorder traversal of a nonempty binary tree is defined as follows:
Visit the root node
Traverse the left sub-tree in preorder <root><left><right>
,_ Traverse the right sub-tree in preorder
8 C
C
fig Binary tree
The preorder traversal output of the given tree is: AB DH IE CF G
The preorder is also known as depth first order.
26

,,, The in-order traversal of a nonen1pty binary tree is defined as follows:
Traverse the left sub-tree in in-order
Visit the root node
- Traverse the right sub-tree in inorder
fig Binary tree
<left><root><right>
C
C
The in-order traversal output ofthe given tree is lf DI BE AF CG
27

,,, 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
- Visit the root node
<left><right><root>
C
D £
fig Binary tree
• • • + � -
The post-order traversal output ofthe given tree is HID E BF G CA
28

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29

t. Binary Search Tree - Used in many search applications that constantly show and hide
data, such as data. For example, map and set objects in many libraries.
2. Binary Space Partition - Used in almost any 3D video game to detennine which objects
need to be rendered.
3. Binary Tries - Used in almost every high-bandwidth router to store router tables.
4. SyntaxTree - Constructed by compilers and (implicit) calculators to parse expressions.
5. Hash Trees - Used in P2P prograrns and special image signatures that require a hash to
be validated, but the entire file is not available.
6. Heaps - Used to implement efficient priority queues and also used in heap sort.
7. Treap - Randomized data stlucture for Vireless networks and mernory allocation.
8. T-Tree - Although most databases use a form of 8-tree to store data on the drive,
databases that store all (most) data often use T-trees.
9. Huffman Coding Tree (Chip Uni) - Used in compression algorithms, eg. For exarnple,
in .jpeg and .tnp3.GGM Trees file formats - used in cryptographic applications to
generate a tree with pseudo-randorn nurnbers.
30

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Chapter 7 - Binary Search Tree in the context of DSA.pdf