Let’s solve the questions step by step based on the binary tree provided:

12. Which node is the root?

The root of the tree is the topmost node, which is ‘-’ (subtraction operator).

13. What are the internal nodes?

Internal nodes are nodes that have at least one child. In this tree, the internal nodes are:

• - (root)
• / (division)
• + (addition)
• / (division below left subtree)
• * (multiplication below right subtree)

So, the internal nodes are: ‘-’, ‘/’, ‘+’, ‘/’, and ‘*’.

14. How many ancestors does the node ‘6’ have?

To find the ancestors of the node labeled ‘6’:

• Start from the root and move upward until you reach the node itself.
• Ancestors of ‘6’ are: ‘-’ (root).

Thus, the node ‘6’ has 1 ancestor.

15. Which nodes are in the subtree rooted at the only ‘/’ node?

The subtree rooted at the only ‘/’ node is located under the root's left child (division operator). The nodes in this subtree are:

• Root of the subtree: ‘/’
• Its children: 9 and 5.

Nodes: ‘/’, ‘9’, and ‘5’.

16. What is the height of the tree?

The height of a tree is the length of the longest path from the root to a leaf node.

• Path from root to 3/1: 3 levels
• Path from root to 6: 2 levels
• Path from root to 7/4: 3 levels

The height of the tree is 3.

17. What is the preorder traversal of the tree?

In preorder traversal, visit nodes in the order: root → left subtree → right subtree.

Preorder traversal:

1. Visit root: ‘-’
2. Traverse left subtree of root: ‘/’ → ‘+’ → ‘3’ → ‘1’ → ‘/’ → ‘9’ → ‘5’
3. Traverse right subtree of root: ‘*’ → ‘2’ → ‘+’ → ‘3’ → ‘7’ → ‘4’ → ‘6’

Result: ‘-’, ‘/’, ‘+’, ‘3’, ‘1’, ‘/’, ‘9’, ‘5’, ‘*’, ‘2’, ‘+’, ‘3’, ‘7’, ‘4’, ‘6’.

18. What is the postorder traversal of the tree?

In postorder traversal, visit nodes in the order: left subtree → right subtree → root.

Postorder traversal:

1. Traverse left subtree: ‘3’, ‘1’, ‘+’, ‘9’, ‘5’, ‘/’, ‘/’
2. Traverse right subtree: ‘2’, ‘3’, ‘7’, ‘4’, ‘+’, ‘*’, ‘6’
3. Visit root: ‘-’

Result: ‘3’, ‘1’, ‘+’, ‘9’, ‘5’, ‘/’, ‘/’, ‘2’, ‘3’, ‘7’, ‘4’, ‘+’, ‘*’, ‘6’, ‘-’.

19. What is the in-order traversal of the tree?

In in-order traversal, visit nodes in the order: left subtree → root → right subtree.

In-order traversal:

1. Traverse left subtree: ‘3’, ‘+’, ‘1’, ‘/’, ‘9’, ‘/’, ‘5’
2. Visit root: ‘-’
3. Traverse right subtree: ‘2’, ‘*’, ‘3’, ‘+’, ‘7’, ‘4’, ‘6’

Result: ‘3’, ‘+’, ‘1’, ‘/’, ‘9’, ‘/’, ‘5’, ‘-’, ‘2’, ‘*’, ‘3’, ‘+’, ‘7’, ‘4’, ‘6’.

20. What is the result of the arithmetic expression, assuming an in-order traversal?

To compute the result:

1. Use the given in-order traversal: ‘3 + 1 / 9 / 5 - 2 * 3 + 7 / 4 - 6’
2. Evaluate step by step using proper operator precedence:
   • Division: $1 / 9 = 0.1111...$, then $0.1111 / 5 = 0.0222...$
   • Multiplication: $2 \times 3 = 6$
   • Addition: $3 + 0.0222 = 3.0222$, then $3.0222 + 7 = 10.0222$
   • Subtraction: $10.0222 - 4 = 6.0222$, then $6.0222 - 6 = 0.0222$.

Result: Approximately 0.02.

Any clarifications or further questions?

Here are five related questions for deeper understanding:

1. What is the difference between preorder, postorder, and in-order traversals in binary trees?
2. How can you determine the height of a binary tree programmatically?
3. What are the key properties of binary trees used in arithmetic expression evaluation?
4. How does operator precedence influence the result of a binary expression tree evaluation?
5. What is a balanced binary tree, and how does it differ from the tree in the diagram?

Tip: In binary tree problems, always focus on traversal patterns and their effect on computations.