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### Q1
Explain why we need to maintain a binary search tree in balance such as an AVL tree.
```
Only a balanced tree can guarantee search performance of O(logN).
and, AVL tree’s theoretical search performance is 1.44 log2N. (see lecture notes)
```
### Q2
Build a binary search tree with the following integers being inserted in the order from
left to right: 72, 48, 35, 80, 60, 85, 75, 21, 9, 28
```
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
35 60 75 85
/
21
/ \
9 28
```
### Q3
With the integers in Q2, show the steps of building an AVL tree. Indicate also the
balance value of each node and the rotations so required to maintain the tree in
balance.
```
insert 72
72
insert 48
72
/
48
insert 35
72
/
48
/
35
=>Right rotate at 72
48
/ \
/ \
35 72
insert 80
48
/ \
/ \
35 72
\
80
insert 60
48
/ \
/ \
35 72
/ \
/ \
60 80
insert 85
48
/ \
/ \
35 72
/ \
/ \
60 80
\
85
=> Left rotate at 48
72
/ \
/ \
48 80
/ \ \
/ \ 85
35 60
insert 75
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
35 60 75 85
insert 21
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
35 60 75 85
/
21
insert 9
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
35 60 75 85
/
21
/
9
=> Right rotate at 35
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
21 60 75 85
/ \
9 35
insert 28
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
21 60 75 85
/ \
9 35
/
28
=> double rotate: left rotate at 21
72
/ \
/ \
/ \
/ \
48 80
/ \ / \
/ \ / \
35 60 75 85
/
21
/ \
9 28
=> double rotate: right ritate at 48
72
/ \
/ \
/ \
/ \
35 80
/ \ / \
/ \ / \
21 48 75 85
/ \ \
9 28 60
```
### Q4
Show the steps of deleting the following nodes from the AVL tree in Q3. Indicate also the balance value of each node and the rotations so required to maintain the tree in balance.
60, 48
```
delete 60
72
/ \
/ \
/ \
/ \
35 80
/ \ / \
/ \ / \
21 48 75 85
/ \
9 28
delete 48
72
/ \
/ \
35 80
/ / \
21 / \
/ \ 75 85
9 28
=> rotate at 35
72
/ \
/ \
/ \
21 80
/ \ / \
9 35 / \
/ 75 85
28
```