Definition of Binary Tree, Introduction to Binary Search Tree, Binary Tree Search Algorithm, Insert a value in BST, Delete a value from BST

1. BINARY SEARCH
TREE
Group leader:
Jethanand
Class: BSCS – 3rd A

2. BINARY SEARCH TREE:
• Binary search tree is a data structure that quickly allows us to maintain a
sorted list of numbers. It is called a binary tree because each tree node has
a maximum of two children. It is called a search tree because it can be used
to search for the presence of a number in O(log(n)) time.

3. STRUCTURE:
• Each node in a BST contains a key/value pair and has
at most two children: a left child and a right child.
• The left child contains a key/value pair smaller than
its parent, while the right child contains a key/value
pair greater than its parent.
• This property holds true for every node in the tree,
making it easy to search for elements.

4. BASIC OPERATIONS
In-order Traversal − Traverses a tree in an in-order manner.
Pre-order Traversal − Traverses a tree in a pre-order manner.
Post-order Traversal − Traverses a tree in a post-order manner.
Search − Searches an element in a tree.
Insert − Inserts an element in a Tree.
Deletion- Delete an element in tree.

5. TEVERSAL IN BINARY SEARCH TREE:
• 1. In order Teversal:
Visit the left subtree, then the current node, and
finally the right subtree. It prints the nodes in ascending order.(L,N,R)
• Example: 3,4,5,7,9,14,15,16,17,18,20
• 2. Preorder Traversal:
Visit the current node, then the left subtree, and
finally the right subtree. (N,L,R)
• Example: 14,4,3,9,7,5,15,18,16,17,20
• 3. Post order Traversal:
Visit the left subtree, then the right subtree, and
finally the current node. (L,R,N)
• Example: 3,5,7,9,4,17,16,20,18,15,14

6. SEARCH OPERATION:
• The key property of a BST enables efficient searching. Starting from the root, we
compare the target value with the current node.
• If the target is smaller, we move to the left child; if it's larger, we move to the right
child.
• We repeat this process until we find a match or reach a null node, indicating that
the value is not present in the tree.
• The search operation has an average and worst-case time complexity of O(log n) in
balanced BSTs, where n is the number of elements.

7. EXAMPLE:
• Consider the graph shown below and the key = 6.
• Step 1: Initially compare the key with the root i.e., 8. As 6
is less than 8, search in the left subtree of 8.
• Step 2: Now compare the key with 3. As key is greater
than 3, search next in the right subtree of 3.
• Step 3:Now compare the key with 6. The value of the key
is 6. So we have found the key.

8. INSERTION OPERATION:
 The insertion logic into BST is similar to its searching operation. A new value is always inserted
at the leaf node of the BST.
 Compare the value with the root of the BST.
 If the value to be inserted is less than the root, move to the left subtree.
 Otherwise, if the value is greater than the root, move to the right subtree.
 Continue this process, until we hit a leaf node.
 If the value is less than the leaf, create a left child of the leaf and insert the value.
 Otherwise, if the value is greater than the leaf, create a right child of the leaf and insert the value
in the right child.

9. EXAMPLE:
• Consider the following BST and the value
= 40 to be added.
• Initially, 40 is less than 100. So move to
the left subtree.
• Now, 40 is greater than 20. So move to
the right subtree.
• Now we reach the leaf node 30. As 40 is
greater than 30, create right child of 30
and insert the value 40.
After insertion
Before insertion

10. DELETION OPERATION
 Deleting a node from a BST involves three possible cases:
 Case 1: The node has no children: We can simply remove the node.
 Case 2: The node has one child: We replace the node with its child.
 Case 3: The node has two children: We find the minimum value in the right subtree (or the maximum value
in the left subtree) and replace the node with it, then delete the duplicate value from the subtree.
 The time complexity of the deletion operation is also O(log n) in balanced BSTs.