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### Q2(c)
Now, perform some experiments to test the searching performance in the two approaches. Record the results in the table below. You are free to try some more words.
| Word to search for | Result (found/ not found) | Time needed (BST) | Time needed (linked list)
| --- | --- | --- | --- |
| water | Found | 2500 | 7100 |
| ever | Found | 4800 | 77600 |
| snail | Not Found | 4500 | 74200 |
| better | Not Found | 3900 | 80700 |
| apple | Found | 5700 | 49700 |
| door | Found | 3300 | 7300 |
| foolish | Found | 4000 | 52000 |
### Q4
A proper binary tree contains N nodes. What is the number of leaf nodes in the tree? What is the
number of non-leaf nodes in the tree?
```
Leaf nodes = N/2+1 (N must be an odd number, so take the integer of N/2.)
Non-leaf nodes = N/2
```
### Q5
A complete binary tree has a depth of 8. What is the total number of nodes in the tree? Give the
relationship of the depth and the total number of nodes of a complete binary tree in a mathematic
expression.
```
Total nodes = 2^(8+1) – 1 = 511
```
### Q6a
Draw the array of the binary tree if it is implemented in an
array.
```
[Q]
/ \
[B] [U]
\ / \
[G] [R][W]
/ \
[E] [J]
\ / \
[F] [I] [P]
/
[H]
```
```
0:Q 1:B 2:U 4:G 5:R 6:W 9:E 10:J 20:F 21:I 22:P 43:H
```
### Q6(b)
(b)List the node sequence by
(i) pre-order traversal
```
Q B G E F J I H P U R W
```
(ii) in-order traversal
```
B E F G H I J P Q R U W
```
(iii) post-order traversal
```
F E H I P J G B R W U Q
```
### Q7
Follow the algorithm outlined in the lecture notes, depict the Binary Search Trees created by inserting the items below from left to right. Perform an in-order traversal to check if the nodes are visited in ascending order.
(a) M J W S G L K A B P Z X Y R T
```
M
/ \
/ \
/ \
/ \
J W
/ \ / \
G L / \
/ / / \
A K S Z
\ / \ /
B / \ X
P T \
\ Y
R
```
(b) P W J Y B L S G K M A Z X T R
```
P
/ \
/ \
/ \
/ \
/ \
/ \
/ \
/ \
J W
/ \ / \
/ \ / \
/ \ / \
B L / \
/ \ / \ S Y
A G / \ / \ / \
K M / \ / \
R T X Z
```
(c) P W J Y B L S G K M A R
```
P
/ \
/ \
/ \
/ \
/ \
J W
/ \ / \
/ \ S Y
/ \ /
B L R
/ \ / \
A G / \
K M
```
(d) P W J Y B L S G K M A
```
P
/ \
/ \
/ \
/ \
/ \
J W
/ \ / \
/ \ S Y
/ \
B L
/ \ / \
A G / \
K M
```
(e) A B C D E F G H I J K L
```
A
\
B
\
C
\
D
\
E
\
F
\
G
\
H
\
I
\
J
\
K
\
L
```
(f) L K J I H G F E D C B A
```
L
/
K
/
J
/
I
/
H
/
G
/
F
/
E
/
D
/
C
/
B
/
A
```
### Q8
Outline an algorithm of searching an item in a Binary Search Tree.
```
BinaryNode search (BinaryNode t, key x)
begin
if t is null
return null;
if (x is less than t.data.key)
return search(t.left, x);
else if (x is greater than t.data.key)
return search(t.right, x);
else
return t;
end
```
### Q10
Give the postfix and prefix expressions as well as the expression tree.
((A + B) * C) / (D + E) * F
```
[*]
/ \
[/] [F]
/ \
/ \
/ \
[*] [+]
/ \ / \
[+] [C] [D] [E]
/ \
/ \
[A] [B]
pre-order traversal on the expression tree to yields the prefix expression
Prefix expression: * / * + A B C + D E F
```