Question

Create an AVL tree with a height of 3 where the root of the tree would be the imbalanced alpha node if another single node X is inserted into the tree. Show the creation of the tree starting with the empty tree. Show each individual step of inserting the nodes and at each step state which rotation was done (if a rotation was done). Each value in the tree must be an integer and unique. For this problem, you can use drawings and even the visualization tools. You don’t need to implement this problem. For your AVL tree in the problem above, now what is the node X which causes the tree to be imbalanced with the alpha node at the root? Show inserting and balancing the tree. What type of imbalance is it?

Answer 1

An AVL tree becomes imbalanced at the root when a node violates the AVL tree balancing rules. We can demonstrate this by creating an AVL tree with a height of 3, and inserting nodes until the tree becomes imbalanced. The imbalanced node at the root can be corrected using the appropriate rotation.

Step-by-step explanation:

When a single node X is inserted in a way that defies the AVL tree balancing guidelines, the result is an imbalanced AVL tree at the root. The difference between a node's left and right subtree heights in an AVL tree is limited to one. To demonstrate this, let's create an AVL tree with a height of 3, starting with an empty tree:

1. Insert node 1 (no rotation)
2. Insert node 2 (no rotation)
3. Insert node 3 (no rotation)
4. Insert node 4 (no rotation)
5. Insert node 5 (left rotation at root)
6. Insert node 6 (right rotation at root)
7. Insert node 7 (right rotation at root)
8. Insert node 8 (no rotation)

After inserting and balancing the tree, the imbalanced alpha node at the root is caused by inserting node 7. This creates a left-right imbalance, which requires a right rotation at the root to restore balance.