1. Tree data structure

2. How should I decide which data structure to use?
What needs to be stored?
Cost of operations
Memory usage
Ease of implementation
What do we use a tree for?
Organizational Hierarchy

6. ROOT
Height of x = no. of edges in longest path from x to a leaf
For example: Height of 3 = 2
Height of root = 3 = height of tree

7. Binary Tree Definition: A tree in which each node can
ROOT
Definition: A tree in which each node can
Have at most 2 children

8. Pointer Tree data Left Childs NULL Right Childs Struct node {
Address of
Right Child
Address of
Left Child
Struct node {
int data;
struct node* left;
struct node* right }
data
Applications:
Storing naturally hierarchical data -> e.g. file system.
Organize data for quick search, insertion, deletion.
Network routing algorithm.
AND MORE…
Left Childs
NULL
Right Childs

9. Binary Tree
Binary Tree, each node can have at most 2 children

10. Strict / Proper Binary Tree
Each node can have either 2 or 0 children

11. Complete Binary Tree
All levels except possibly the last are completely filled, and all nodes are as left as possible
Maximum number of nodes at any level (i) is 2i

12. Perfect Binary Tree =2No. of levels-1 Maximum number of nodes
Maximum number of nodes in a binary tree with height (h)
= …….+2h
=2h+1-1
=2No. of levels-1
Maximum number of nodes
= 24-1 = 15

13. Perfect Binary Tree
A complete binary tree in which all levels are full
Number of nodes will be maximum for a height

14. Binary Search Tree Height of a perfect binary tree with n nodes
Now, if n = 15 then
h = 𝑙𝑜𝑔 −1 0
= 𝑙𝑜𝑔 2 16 – 1 = 4 – 1 = 3
Height : number of edges in longest path from root to a leaf
Height of an empty tree = -1
Height of a tree with 1 node = 0

15. We can implement binary tree using: Dynamically created nodes
struct node {
int data;
struct node *left;
Struct node *right; };
b) Arrays
root
But how to store information about the links
For node at index c in a complete binary tree
Left-child index = 2i
Right-childe index = 2i +2 2 4 6 8 2 4 1 5 8 7 9 2 4 1 5 8 7 9

16. Binary Search Tree
What data structure will you use to store a modifiable collection of data?
We want to be able to perform the following operations:
search (x) //search for an item x
insert(x) // insert an item x
remove(x) // delete an item x
Choices: array, linked list….

17. Array (unsorted) search(x): O(n) end insert(x): O(1) insert(7) remove(x): O(n) end remove(3) Linked list: search(x): O(n) insert(x): O(1) //at the head remove(x): O(n) head 1 3 5 7 1 5 7 3 5 7

18. We can perform binary search in a sorted array in O(logn) end Array (sorted array) Search(x) : O(logn) sorted array Insert(x): O(n) end Remove(x): O(n) insert(6) BST (Balanced)For n records, log n comparisons if 1 comparison = 10 −6 𝑠𝑒𝑐 Search(x): O(logn) n = 2 30 = 30 x 10 −6 𝑠𝑒𝑐 // 2 30 ≈ 1 billion Insert(x): O(logn) Remove(x): O(logn) 3 5 7 9 3 5 6 7 9

19. Binary Search Tree (BST)
root
A Binary Tree in which for each node, the value of all the nodes in left subtree is lesser or equal and the value of all the nodes in the right subtree is greater.
BST
BST
root
Right-Subtree (greater)
Left-Subtree (lesser or equal) 15 9 19
root 7 11 16 24
BST?

20. So how do we achieve O(log n) in BST for search, insert, and remove?
start
end
end new 7 9 11 15 16 19 24
Search(9)
Search Space Reduction Process
BST n
n/2
log _2 n steps
n/4
n/2^k = 1 . 1
2^k = n
Search(11)
k = log_2 n

21. root BST (unbalanced) root BST (unbalanced) 24 Search Space Reduction19
root
O(log n)
(avg. case) 16 7 15 9
O(n)
(worst case) 11
BST (unbalanced) 11 9 15 7 16 19 24

22. Insert (18)
O(log n)
Similarly delete() is going to take O(log n)