https://file.aiquickdraw.com/mathbot/file/173190676506061wxwc2p.jpg

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).

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### 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 ‘*’.**

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### 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.**

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### 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’.**

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### 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**.

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### 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’.**

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### 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’, ‘-’.**

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### 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’.**

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### 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**.

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### 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.

Questions about a binary tree representing an arithmetic expression with various traversal methods and evaluations.

Explore the properties of a binary tree that represents an arithmetic expression, including internal nodes, traversal methods, and the evaluation of expressions. Suitable for high school students.

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