https://r2.erweima.ai/mathbot/image/1718703056973emfgd1lc.jpg

The image contains two parts related to binary search and binary search trees. Below is the translated and interpreted text along with the tasks requested:

### Part 1
**Given the ordered list [3, 4, 5, 7, 24, 30, 42, 54, 63, 72, 87, 95], perform a binary search and answer the following questions:**

1. Draw the decision tree for the binary search process.
2. When searching for element 54, which elements are compared?
3. When searching for element 90, which elements are compared?
4. Assuming the search probability for each element is equal, find the average search length when the search is successful.

### Part 2
**In an initially empty binary search tree, insert the sequence of keys [12, 7, 17, 11, 16, 2, 13, 9, 21, 4]. Draw the resulting binary search tree.**

---

### Answers and Explanations

#### Part 1

1. **Decision Tree for Binary Search:**
   To draw the decision tree, we follow the binary search steps. Here is the process broken down:
   
   - Initial list: [3, 4, 5, 7, 24, 30, 42, 54, 63, 72, 87, 95]
   - Mid element: 42 (index 6)
   - Left sub-list: [3, 4, 5, 7, 24, 30]
   - Right sub-list: [54, 63, 72, 87, 95]
   - Continue the process for each sub-list.
   
   This will result in a decision tree where each node represents a comparison with the middle element, and branches represent the left and right sub-lists.

2. **Search for element 54:**
   - Compare with 42 (54 > 42, move right)
   - Compare with 72 (54 < 72, move left)
   - Compare with 54 (found)
   
   Elements compared: 42, 72, 54.

3. **Search for element 90:**
   - Compare with 42 (90 > 42, move right)
   - Compare with 72 (90 > 72, move right)
   - Compare with 87 (90 > 87, move right)
   - Compare with 95 (90 < 95, move left)
   - Element not found.
   
   Elements compared: 42, 72, 87, 95.

4. **Average Search Length:**
   The average search length for a successful search can be calculated by considering the depth of each element in the decision tree. For the given ordered list, let's assume equal probability for each element:
   - Depths: 3(4), 4(4), 5(4), 7(4), 24(4), 30(4), 42(3), 54(3), 63(3), 72(3), 87(3), 95(3)
   - Average = (3+4+4+4+4+4+3+3+3+3+3+3)/12 = 3.5
   
   Thus, the average search length is 3.5 comparisons.

#### Part 2

**Binary Search Tree Insertion:**

- Insert 12: root (12)
- Insert 7: left child of 12
- Insert 17: right child of 12
- Insert 11: right child of 7
- Insert 16: left child of 17
- Insert 2: left child of 7
- Insert 13: left child of 16
- Insert 9: left child of 11
- Insert 21: right child of 17
- Insert 4: right child of 2

The resulting binary search tree is:

```
         12
        /   \
      7     17
     / \    / \
    2  11  16  21
     \  /    \
      4 9    13
```

Would you like more details or have any questions about the processes described? 

### Suggested Questions:

1. How does binary search improve search efficiency compared to linear search?
2. Can you explain the process of inserting elements into a binary search tree?
3. What is the time complexity of binary search?
4. How is the average search length in a binary search tree calculated?
5. What are the applications of binary search trees in computer science?

### Tip
When performing binary search, always ensure the list is sorted before starting the search process.

Explore binary search and binary search trees with a detailed explanation and solutions. Learn about decision trees in binary search and the process of inserting keys into a binary search tree. Suitable for undergraduate students and those interested in algorithms and data structures.

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