Suppose
T is a
binary tree

Then T is
called binary search tree if each node N of
T
has the following property

The
value at N is greater
than every value in the left sub-tree of N and is less than
every value in the right sub-tree of N

### Binary Search Tree (BST)

### Two binary search trees representing the same set:

Average
depth of a node is O(log N); maximum depth of a node is O(N)

## Searching and Inserting in BST

Algorithm
to find the location of ITEM
in the BST T
or
insert ITEM
as
a new node in its appropriate place in the tree

[a]
Compare ITEM with the root node N of the
tree

(i)
If ITEM < N, proceed to the left child of N

(ii)
If ITEM > N, proceed to the right child of N

[b]
Repeat Step (a) until one of the following occurs

(i)
We meet a node N such that ITEM = N. In this case search is successful

(ii)
We meet an empty sub-tree, which indicates that search is unsuccessful and we
insert ITEM in place of empty subtree

### Searching BST

•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.

**Example :**

### Example demo :

### Locating an ITEM

A binary search tree T is in memory
and an ITEM of information is given. This procedure finds the location LOC of
ITEM in T and also the location of the parent PAR of ITEM

Three
special cases:

[1]
LOC == NULL and PAR == NULL, tree is empty

[2]
LOC ¹
and PAR == NULL, ITEM is the root of T

## ALGORIthm :

[1]
[Tree empty ?]

If ROOT == NULL, then

Set LOC = NULL, PAR = NULL, Exit

[2]
[ITEM at root ?]

If ROOT->INFO == ITEM, then

Set LOC = ROOT, PAR = NULL, Exit

[3]
[Initialize pointer PTR and SAVE]

If ITEM < ROOT->INFO then

Set PTR = ROOT->LEFT, SAVE = ROOT

Else

Set PTR = ROOT->RIGHT, SAVE =ROOT

[4]
Repeat Step 5 and 6 while PTR ¹
NULL

[5]
[ITEM Found ?]

If ITEM == PTR->INFO, then

Set LOC = PTR, PAR = SAVE, Exit

[6] If ITEM < PTR->INFO, then

Set SAVE = PTR, PTR = PTR->LEFT

Else

Set SAVE = PTR, PTR = PTR->RIGHT

[7]
[Search Unsuccessful]

Set LOC = NULL, PAR = SAVE

[8]
Exit

A
binary search Tree T is in memory and an ITEM of information is given.
Algorithm to find the location LOC of ITEM in T or adds ITEM as a new node in T
at location LOC.

[1]
Call FIND(INFO, LEFT,RIGHT,ROOT,ITEM,LOC,PAR)

[2]
If LOC ¹
NULL, then Exit

[3]
[Copy the ITEM into the node NEW]

(a) Create a node NEW

(b) NEW->INFO = ITEM

( c) Set LOC = NEW, NEW->LEFT = NULL,
NEW->RIGHT = NULL

[4]
[Add ITEM to tree]

If PAR = NULL

Set ROOT = NEW

Else

If ITEM < PAR->INFO, then

Set PAR->LEFT = NEW

Else

Set PAR->RIGHT = NEW

[5]
Exit

Insert Algorithm

• If value we want to insert < key
of current node, we have to go to the left subtree

• Otherwise we have to go to the
right subtree

•If the current node is empty (not
existing) create a node with the value we are inserting and place it here.

## Deletion from BST

T is
a binary tree. Delete an ITEM from the tree T

Deleting
a node N
from tree depends primarily on the number of children of node N

There
are three cases in deletion

Case
1.
N has no children. N is deleted from the T by replacing the location of N
in the parent node of N by null pointer

Case
2.
N has exactly one children. N is deleted from the T by simply replacing the
location of N in the parent node of N by
the location of the only child of N

### Types of Binary Trees

•Degenerate – only one child

•Balanced – mostly two children

•Complete – always two children

Binary Trees Properties

–Degenerate

•Height = O(n) for n nodes

•Similar to linear list

-Balanced

–Height = O( log(n) ) for n nodes

–Useful for searches

•Key property Of BST

–Value at node

•Smaller values in left subtree

•Larger values in right subtree

–Example

•X > Y

•X < Z

### Binary Search Properties

•Time of search

–Proportional to height of tree

–Balanced binary tree

•O( log(n) ) time

–Degenerate tree

•O( n ) time

•Like searching linked list /
unsorted array

•Requires

–Ability to compare
key values

### Binary Search Tree Construction

•How to build & maintain binary
trees?

–Insertion

–Deletion

•Maintain key property (invariant)

–Smaller values in left subtree

–Larger values in right subtree

### Binary Search Tree – Insertion

•Algorithm

Perform
search for value X

Search
will end at node Y (if X not in tree)

If X
< Y, insert new leaf X as new left subtree
for Y

If X
> Y, insert new leaf X as new right subtree
for Y

•Observations

–O( log(n) ) operation for balanced
tree

–Insertions may unbalance tree

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