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.