2. What is a Binary Search Tree?
PROPERTY 1:
Each node can have upto two successor nodes.
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3. What is a Binary Search Tree?
PROPERTY 2:
A unique path exists from the root to every node.
This is not valid binary tree!
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4. Some Basic Information
The successor nodes of a node are called its children.
The predecessor node of a node is called its parent.
The first node is called the root.
A node without a child is called a leaf.
Nodes with the same parent are called siblings.
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5. Basic Information(Cont.)
Nodes are organized in levels.indexed from 0.
Level(or depth) of a node is the number of edges in the path
from the root to that node.
Height of a tree (h):No. of levels (L)
Full tree:Every node has exactly two children and all the leaves
are on the same level.
Complete tree: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.
Full Tree – Complete Tree
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6. Binary Search Tree Property
-
A binary search tree is a rooted binary tree, whose internal nodes
each store a key (and optionally, an associated value) and each have
two distinguished sub-trees, commonly denoted left and right. The
tree additionally satisfies thebinary search tree property, which
states that the key in each node must be greater than all keys
stored in the left sub-tree, and smaller than all keys in right
sub-tree
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7. Max No.of Nodes
Max no. of nodes at a level l is 2l
.
Total no. of nodes N of a full tree with height h = 2h
− 1.
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8. Height h AND It’s Importance
Since, N = 2h
− 1
=> 2h
= N + 1
=> h = log(N + 1)
Hence, h -> O(log(N))
Tree operations (e.g., insert, delete, retrieve etc.)are typically
expressed in terms of h.
So, h determines running time!
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9. Search In a Binary Tree
Start at the root
Search the tree level by level, until you find the element you are
searching for or you reach a leaf.
Is this better than searching a linked list?
NO! -> O(N)
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10. Search In A Binary Search Tree
1 Start at the root
2 Compare the value of the item you are searching for with the
value stored at the root
3 If the values are equal, then item found; otherwise, if it is a leaf
node, then not found
4 If it is less than the value stored at the root, then search the left
subtree
5 If it is greater than the value stored at the root, then search the
right subtree
6 Repeat steps 2-6 for the root of the subtree chosen in the
previous step 4 or 5.
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11. Search in BST(Cont.)
Is this better than searching a linked list?
YES-> O(log(N))
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12. Function for No. of Nodes
Recursive implementation
No. of nodes in a tree
= No. of nodes in left subtree + No. of nodes in right
subtree+1
What is the size factor?
Number of nodes in the tree we are examining
What is the base case?
The tree is empty
What is the general case?
CountNodes(Left(tree)) + CountNodes(Right(tree)) +1
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13. Function RetrieveItem()
What is the size of the problem?
Number of nodes in the tree we are examining
What is the base case(s)?
1 When the key is found
2 The tree is empty (key was not found)
What is the general case?
Search in the left or right subtrees
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15. Function InsertItem()
What is the size of the problem?
Number of nodes in the tree we are examining
What is the base case(s)?
The tree is empty
What is the general case?
Choose the left or right subtree
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17. Function DeleteItem()
First, find the item.
Then, delete it.
Binary search tree property must be preserved!!
We need to consider three different cases
1 Deleting a leaf
2 Deleting a node with only one child
3 Deleting a node with two children
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21. Deleting a node with two children(Cont.)
Find predecessor (i.e., rightmost node in the left subtree)
Replace the data of the node to be deleted with predecessor’s
data
Delete predecessor node
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22. Function DeleteItem()
What is the size of the problem?
Number of nodes in the tree we are examining
What is the base case(s)?
Key to be deleted was found
What is the general case?
Choose the left or right subtree
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23. Traversals
There are mainly three ways to traverse a tree:
1 Inorder Traversal
2 Postorder Traversal
3 Preorder Traversal
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25. Print the Tree
For printing the Binary Search Tree in sorted order we can use
Inorder Traversal.
Result: A D J M Q R T
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26. Use Of A Binary Search Tree
In comparison to other trees using a Binary Search Tree we can
efficiently perform the following:
Search for a particular element.
Sorting of elements.
Find Maximum/Minimum.
Delete a particular element.
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