Binary search trees (BSTs) are data structures that allow for efficient searching, insertion, and deletion. Nodes in a BST are organized so that all left descendants of a node are less than the node's value and all right descendants are greater. This property allows values to be found, inserted, or deleted in O(log n) time on average. Searching involves recursively checking if the target value is less than or greater than the current node's value. Insertion follows the search process and adds the new node in the appropriate place. Deletion handles three cases: removing a leaf, node with one child, or node with two children.
Binary search works on sorted arrays. Binary search begins by comparing an element in the middle of the array with the target value. If the target value matches the element, its position in the array is returned. If the target value is less than the element, the search continues in the lower half of the array.
Traversal is a process to visit all the nodes of a tree and may print their values too. Because, all nodes are connected via edges (links) we always start from the root (head) node. That is, we cannot randomly access a node in a tree.
Presentation On Binary Search Tree using Linked List Concept which includes Traversing the tree in Inorder, Preorder and Postorder Methods and also searching the element in the Tree
Binary search works on sorted arrays. Binary search begins by comparing an element in the middle of the array with the target value. If the target value matches the element, its position in the array is returned. If the target value is less than the element, the search continues in the lower half of the array.
Traversal is a process to visit all the nodes of a tree and may print their values too. Because, all nodes are connected via edges (links) we always start from the root (head) node. That is, we cannot randomly access a node in a tree.
Presentation On Binary Search Tree using Linked List Concept which includes Traversing the tree in Inorder, Preorder and Postorder Methods and also searching the element in the Tree
Content of slide
Tree
Binary tree Implementation
Binary Search Tree
BST Operations
Traversal
Insertion
Deletion
Types of BST
Complexity in BST
Applications of BST
This PPT is all about the Tree basic on fundamentals of B and B+ Tree with it's Various (Search,Insert and Delete) Operations performed on it and their Examples...
Content of slide
Tree
Binary tree Implementation
Binary Search Tree
BST Operations
Traversal
Insertion
Deletion
Types of BST
Complexity in BST
Applications of BST
This PPT is all about the Tree basic on fundamentals of B and B+ Tree with it's Various (Search,Insert and Delete) Operations performed on it and their Examples...
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2. Binary Search Trees (BST)
A data structure for efficient searching, insertion
and deletion
Binary search tree property
For every node X
All the keys in its left
subtree are smaller than
the key value in X
All the keys in its right
subtree are larger than the
key value in X
4. Binary Search Trees
Average depth of a node is O(log N)
Maximum depth of a node is O(N)
The same set of keys may have different BSTs
5. Searching BST
If we are searching for root (15), then we are done.
If we are searching for a key < root , then we
should search in the left subtree.
If we are searching for a key > root, then we should
search in the right subtree.
6.
7. Searching (Find)
FIND(info, left, right, root, item, loc, par)- finds the item in tree T with root is root and
info, left and right is three array represented in memory. This algorithm returns loc
i.e. location of item and par i.e. parent.
1. [Tree Empty??]
if root==NULL, then set LOC=NULL & PAR=NULL and return.
1. [Item root ??]
If item==INFO[ROOT], then LOC=ROOT & PAR=NULL and return.
1. [Initialize pointer ptr and save]
If item<INFO[ROOT]
then set PTR = LEFT[ROOT] and SAVE=ROOT
Else
set PTR = RIGHT[ROOT] and SAVE=ROOT
[End of if]
1. Repeat 5 and 6 while ptr!=NULL
2. [item found??]
If ITEM=INFO[PTR], then set LOC=PTR and PAR=SAVE, and return.
1. If ITEM<INFO[PTR], then SAVE=PTR and PTR=LEFT[PTR]
Else
Set SAVE=PTR and PTR=RIGHT[PTR]
1. [Search unsuccessful] Set, LOC=NULL and PAR = SAVE
2. Exit
Time complexity: O(height of the tree)
8. Sorting: Inorder Traversal of BST
Inorder Traversal of BST prints out all the keys in
sorted order
Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20
9. Insertion
Proceed down the tree as you would with a find
If X is found, do nothing (or update something)
Otherwise, insert X at the last spot on the path traversed
Time complexity = O(height of the tree)
10. Inserting (ADD node)INSBST(info, left, right, root, item, loc, avail)- insert the item in tree
T with root is root and info, left and right is three array
represented in memory. This algorithm returns loc i.e. location of
item or ADD item as new node in tree.
1. Call FIND(INFO, LEFT, RIGHT, ROOT, ITEM, LOC, PAR)
2. If LOC!=NULL, then Exit.
3. [Copy ITEM into new node in AVAIL list]
a) If AVAIL==NULL, Print “OVER FLOW”;
b) Set NEW=AVAIL, AVAIL=LEFT[AVAIL] and
INFO[NEW]=ITEM.
c) Set LOC=NEW,LEFT[NEW]=RIGHT[NEW]=NULL
4. [ADD ITEM to TREE]
If PAR=NULL then, Set ROOT=NEW.
Else IF ITEM<INFO[PAR] , Set LEFT[PAR]=NEW
Else Set RIGHT[PAR]=NEW
1. Exit
Time complexity: O(height of the tree)
11. Deletion
When we delete a node, we need to consider how we
take care of the children of the deleted node.
This has to be done such that the property of the
search tree is maintained.
12. Deletion under Different Cases
Case 1: the node is a leaf
Delete it immediately
Case 2: the node has one child
Adjust a pointer from the parent to bypass that node
13. Deletion Case 3
Case 3: the node has 2 children
Replace the key of that node with the minimum element
at the right subtree
Delete that minimum element
Has either no child or only right child because if it has a left
child, that left child would be smaller and would have been
chosen. So invoke case 1 or 2.
Time complexity = O(height of the tree)
14. Deletion Algorithm
DEL(INFO, LEFT, RIGHT, ROOT, AVAIL, ITEM)
A binary search tree T is in memory, and an ITEM of information is
given. This algorithm delete ITEM from the tree.
1. Call FIND(INFO, LEFT, RIGHT, ROOT, ITEM, LOC, PAR)
2. If LOC=NULL, then write ITEM not in tree and Exit
3. If RIGHT[LOC]!=NULL and LEFT[LOC]!=NULL, then:
Call CASEB(INFO, LEFT, RIGHT, ROOT, LOC, PAR)
Else:
Call CASEA(INFO, LEFT, RIGHT, ROOT, LOC, PAR)
4. Set LEFT[LOC]:=AVAIL and AVAIL :=LOC.
5. Exit
15. CASEA: only one or, no child
CASEA(INFO, LEFT, RIGHT, ROOT, LOC, PAR)-delete
the Node N at location LOC, where N doesn’t have two
Children. PAR is location of parent node or, PAR=NULL
i.e. ROOT node.
1. [initialize CHILD]
If LEFT[LOC]=NULL and RIGHT[LOC]=NULL, then
CHILD=NULL
Else if LEFT[LOC]!=NULL , then CHILD=LEFT[LOC]
Else CHILD=RIGHT[LOC]
1. If PAR != NULL then: (i.e. NOT A ROOT NODE)
If LOC=LEFT[PAR], then set LEFT[PAR]=CHILD
Else RIGHT[PAR]=CHILD
[End of IF]
Else set ROOT=CHILD.
[End of IF]
1. Exit
16. CASEB: has 2 children
CASEB(INFO, LEFT, RIGHT, ROOT, LOC, PAR)-delete the Node N at location
LOC, where N has two Children. PAR is location of parent node or, PAR=NULL i.e.
ROOT node. SUC gives location of inorder successor and PARSUC gives location
of parent of inorder successor .
1. [Find SUC and PARSUC]
a) Set PTR=RIGHT[LOC] and SAVE=LOC
b) Repeat while LEFT[PTR]!=NULL
Set, SAVE=PTR and PTR=LEFT[PTR]
[END OF LOOP]
a) Set SUC=PTR and PARSUC=SAVE.
2. [Delete SUC] Call CASEA(INFO, LEFT, RIGHT, ROOT, SUC,PARSUC)
3. [replace node N by SUC]
a) If PAR != NULL then: (i.e. NOT A ROOT NODE)
If LOC=LEFT[PAR], then set LEFT[PAR]=SUC
Else RIGHT[PAR]=SUC
[End of IF]
Else set ROOT=SUC.
[End of IF]
b) Set, LEFT[SUC]=LEFT[LOC] and
RIGHT[SUC]=RIGHT[LOC]
4. Exit