https://file.aiquickdraw.com/mathbot/file/17372098917040chcqhw8.jpg

To solve this problem, we analyze the **worst-case running time for inserting \( n \) elements into an initially-empty binary search tree (BST)**.

---

### **Worst-Case Analysis**:
1. In a **binary search tree (BST)**, insertion takes time proportional to the height of the tree.
2. In the **worst case**, the BST becomes a skewed tree (like a linked list) if the elements are inserted in sorted order. In this scenario:
   - The height of the tree is \( n \) for \( n \) elements.
   - Each insertion will require traversing up to \( n-1 \) nodes to find the correct position.

Thus, the **time complexity for inserting \( n \) elements in the worst-case** is:

\[
T(n) = O(n^2)
\]

---

### **Correct Answer**:
The correct choice is:
**a. \( T(n) = O(n^2) \)**

Would you like further details or examples? 😊

---

### **Related Questions**:
1. What is the average-case running time for inserting \( n \) elements into a BST?
2. How does balancing (e.g., AVL or Red-Black Trees) affect the insertion complexity?
3. Why is \( T(n) = O(n \log n) \) incorrect for unbalanced BSTs?
4. What is the best-case scenario for inserting \( n \) elements into a BST?
5. Can we improve BST insertion to achieve \( O(\log n) \) in all cases?

### **Tip**:
To ensure \( O(n \log n) \) insertion, use a **self-balancing binary search tree** like an **AVL tree** or **Red-Black Tree**.

What is the worst-case running time T(n) for inserting n elements into an initially-empty binary search tree?

Understand the worst-case time complexity for inserting n elements into an initially-empty binary search tree. Explore how unbalanced BSTs result in T(n) = O(n^2), with detailed analysis and examples. Perfect for Computer Science students learning about data structures.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation