Consider how maintaining the BST property can help in efficiently splitting and joining trees while keeping the height logarithmic.

Explore the use of auxiliary data structures or metadata (e.g., subtree sizes) to guide the split/join operations and ensure O(log n) time complexity.

Investigate how balancing techniques (e.g., AVL or Red-Black tree rotations) can be adapted to support split and join operations without violating the BST property or increasing the height disproportionately.