Explanation:
a) To insert the numbers 18,3,4,2,8,9,6,1,15,17 into an initially empty red-black tree, we can follow these steps:
1. Insert 18 as the root node of the tree.
- The color of the root node is black.
```
18(B)
/ \
NIL(T) NIL(T)
```
2. Insert 3 as the left child of 18.
- The color of the new node is red.
- Since the parent of 3 is black, the tree remains balanced.
```
18(B)
/ \
3(R) NIL(T)
/ \
NIL(T) NIL(T)
```
3. Insert 4 as the right child of 3.
- The color of the new node is red.
- Since the parent of 4 is red, we need to perform a rotation and recolor the nodes to maintain the red-black tree properties.
```
18(B)
/ \
4(R) 3(R)
/ \
NIL(T) NIL(T)
```
4. Insert 2 as the left child of 3.
- The color of the new node is red.
- Since the parent of 2 is red, we need to perform a rotation and recolor the nodes to maintain the red-black tree properties.
```
18(B)
/ \
4(B) 2(R)
/ \
3(R) NIL(T)
```
5. Insert 8 as the right child of 4.
- The color of the new node is red.
- Since the parent of 8 is black, the tree remains balanced.
```
18(B)
/ \
4(B) 2(R)
/ \
3(R) 8(R)
```
6. Insert 9 as the right child of 8.
- The color of the new node is red.
- Since the parent of 9 is red, we need to perform a rotation and recolor the nodes to maintain the red-black tree properties.
```
18(B)
/ \
4(B) 2(R)
/ \
3(R) 9(R)
\
8(R)
```
7. Insert