Question

Given the following numbers (in order of insertion), depict the resulting AVL Tree.

{42,57,51,21,10,77,88,54,55}
Note: You can use slashes (/ V) to draw a tree in the Canvas textbox. Alternatively, you can depict levels of the tree in rows, using N for places where there are no nodes (see Question 1 above for an example).

Answer 1

To create the AVL Tree from the given numbers, each number is inserted in sequence and the tree is balanced according to AVL rules, where the height difference between left and right subtrees of any node does not exceed one. The final tree structure is provided.

Step-by-step explanation:

The student's question involves constructing an AVL Tree from a given set of numbers. An AVL Tree is a self-balancing binary search tree, where the difference of heights of left and right subtrees cannot be more than one for all nodes. Here's the step-by-step process of inserting the given numbers into the AVL Tree and the resulting tree structure.

• Insert 42 - no balancing needed, this is the root.
• Insert 57 - no balancing needed, it goes to the right of 42.
• Insert 51 - no balancing needed, it goes to the left of 57.
• Insert 21 - no balancing needed, it goes to the left of 42.
• Insert 10 - no balancing needed, it goes to the left of 21.
• Balance checked after every insertion.
• Insert 77 - this leads to right-heavy situation on 57 - rotating 57 left.
• Insert 88 - no balancing needed, it goes to the right of 77.
• Insert 54 - this requires rebalancing - a right rotation on 42.
• Insert 55 - this leads to a left-right case on 54 requiring a left rotation on 51 followed by a right rotation on 54.

The resulting AVL Tree will be:

54
/ \
42 57
/ \ / \
21 51 55 77
/ \ / \
10 N N 88

This tree is balanced according to AVL rules with each node having a height difference of not more than 1 between its left and right subtree.