Introduction

AVL trees are a type of self-balancing binary search tree. Named after their inventors, Adelson-Velsky and Landis, these trees maintain the property that the heights of the left and right subtrees of any node can differ by at most one. This property ensures efficient operations such as insertion, deletion, and lookup, with all operations providing $O(\log n)$ time complexity.

In some applications, it may be necessary to concatenate, merge, or join two AVL trees into a single AVL tree. The process ensures that the resulting tree maintains the AVL property. This task can be more intuitively broken down into operations that respect the tree structure and balance.

Reasons for Concatenation

Before delving into the process, it is important to understand why one might want to concatenate two AVL trees:

1. Data Aggregation: Systems that store and retrieve sorted data can benefit from the quick merging of these data structures.
2. System Scalability: In systems dealing with partitioned data, combining sub-trees can result in efficient data management as systems scale.
3. Range Queries: In scenarios requiring ordered data for range queries, merged AVL trees serve as viable structures.

Pre-requisites

Before concatenating two AVL trees, the following pre-conditions must be satisfied:

• Both trees must be AVL trees individually.
• All keys in the first AVL tree must be less than all keys in the second AVL tree, ensuring the binary search tree properties post-concatenation.

Algorithm Explanation

High-Level Steps

1. Identify the Largest Node in Tree1:
   • This will be the joining node and should effectively become the root after merging.
2. Remove the Largest Node from Tree1:
   • Temporarily remove this node to ease the merge process.
3. Insert Tree2 into Tree1:
   • Integrate the entire second tree while maintaining the AVL property by ensuring balance factors are adjusted.
4. Restore Balance via Rotations and Rebalancing:
   • Ensure that no node in the newly formed tree violates the heights difference property of AVL trees.

Detailed Steps

Step 1: Find the Maximum Node

• Traverse the rightmost path of Tree1 to locate the largest value node.

Step 2: Remove the Maximum Node

• Adjust pointers and balance factors to remove the node, maintaining the AVL properties.

Step 3: Merge the Trees

• Attach Tree2 as a child of the previously extracted node.
• Use rotations where necessary to ensure that the AVL balance is maintained.

Step 4: Rotations and Rebalancing

• Single Right Rotation ($RR$): Suitable for left-heavy trees.
• Single Left Rotation ($LL$): Used for right-heavy trees.
• Left-Right Rotation ($LR$): Used when the left subtree is right-heavy.
• Right-Left Rotation ($RL$): Used when the right subtree is left-heavy.

Each rotation ensures the tree remains balanced by adjusting the sub-nodes.

Example

Suppose we have two AVL trees:

• Tree1: Pre-order traversal results [10, 5, 8]
• Tree2: Pre-order traversal results [15, 12, 18]

To merge:

1. Largest Node in Tree1: Node 10.
2. Remove Node 10, temporarily creating `[5, 8]`.
3. Merge: Integrate Tree2 with `[15, 12, 18]` beneath `[5, 8]` following the BST property.
4. Rebalance if necessary.

The result should look like a balanced AVL tree with the node 10 potentially being reinserted if needed during the rebalance.

Key Points Summary

Conclusion

Concatenating AVL trees involves careful management of tree properties, ensuring the resultant tree retains its self-balancing nature. Understanding rotations and the binary search property is crucial for achieving efficient and effective tree concatenation. The operations hinge on the fundamental ability of AVL trees to accommodate dynamic data while maintaining ordered access and modification speeds. By adhering to these processes, effective data structures can be built on top of AVL trees catering to a variety of computational needs.