https://file.aiquickdraw.com/mathbot/file/1726737175681o1qzb9wm.jpg

To solve the problem, we need to construct a binary search tree (BST) by inserting the numbers in the given order: 10, 1, 3, 5, 15, 12, 16. The height of the tree is defined as the longest path from the root to a leaf node.

### Step-by-Step Insertion:
1. **Insert 10**: This becomes the root of the tree.
   
   ```
   10
   ```

2. **Insert 1**: Since 1 is less than 10, it becomes the left child of 10.

   ```
     10
    /
   1
   ```

3. **Insert 3**: Since 3 is greater than 1 but less than 10, it becomes the right child of 1.

   ```
     10
    /
   1
    \
     3
   ```

4. **Insert 5**: Since 5 is greater than 3 but less than 10, it becomes the right child of 3.

   ```
     10
    /
   1
    \
     3
      \
       5
   ```

5. **Insert 15**: Since 15 is greater than 10, it becomes the right child of 10.

   ```
     10
    /  \
   1    15
    \
     3
      \
       5
   ```

6. **Insert 12**: Since 12 is less than 15 but greater than 10, it becomes the left child of 15.

   ```
     10
    /  \
   1    15
    \   /
     3 12
      \
       5
   ```

7. **Insert 16**: Since 16 is greater than 15, it becomes the right child of 15.

   ```
     10
    /  \
   1    15
    \   / \
     3 12 16
      \
       5
   ```

### Determining the Height:
The height of the binary search tree is the length of the longest path from the root to a leaf node. By examining the tree, the longest path is:
- 10 → 1 → 3 → 5

This path has **4 nodes**, so the height of the tree is **3** (since height is the number of edges, and there are 3 edges along this path).

### Final Answer:
The height of the binary search tree is **3**.

Would you like more details or have any questions?

Here are some related questions to expand on this concept:
1. What are the properties of a binary search tree?
2. How does the order of insertion affect the height of a binary search tree?
3. What is the difference between height and depth in a tree?
4. How would the height change if we inserted numbers in ascending order?
5. Can a binary search tree have a height of zero?

**Tip:** The balance of a binary search tree can significantly impact its height, and techniques like AVL or Red-Black trees can ensure a more balanced structure.

The following numbers are inserted into an empty binary search tree in the given order: 10, 1, 3, 5, 15, 12, 16. What is the height of the binary search tree (the height is the maximum distance of a leaf node from the root)?

Calculate the height of a binary search tree created by inserting the numbers 10, 1, 3, 5, 15, 12, and 16 in order. The tree structure and insertion steps are provided, along with the final answer: height is 3. Suitable for Grades 9-12.

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