1. **State the problem:** We need to construct a splay tree with height 6 containing 10 elements, with the element 7 at level 6. Then, perform a split operation at the value 7. 2. **Constructing the splay tree:** A tree with height 6 means the longest root-to-leaf path has 6 edges (levels numbered from root at 1). We want 10 elements, numbered 1 through 10 for convenience. We arrange so that element 7 is at level 6. For example: - Root is 1 (level 1) - Right child 2 (level 2) - Right child 3 (level 3) - Right child 4 (level 4) - Right child 5 (level 5) - Right child 7 (level 6) - Left child of 7 is 6 - Right child of 7 is 8, and then 9 and 10 as right descendants This makes the height 6 with 7 at level 6. 3. **Visual approximation:** 1 \ 2 \ 3 \ 4 \ 5 \ 7 / \ 6 8 \ 9 \ 10 4. **Performing the split at 7:** The split operation splits the tree into two trees: elements less than 7 and elements greater than or equal to 7. In splay trees, this is typically done by splaying the node with key 7 to the root. **Steps to splay 7 to root:** - Starting at node 7, perform successive zig rotations: - Zig-zig rotations between 5 and 7, then between 3 and 5, and so on until 7 reaches the root. Detailed splaying steps: - Step 1: Zig-zig at 5 and 7 - Step 2: Zig-zig at 3 and 5 - Step 3: Zig-zig at 1 and 3 After splaying, 7 is the root. 5. **Resulting tree:** Root 7 with left subtree of {1,2,3,4,5,6} and right subtree of {8,9,10}. 6. **Split:** - The left tree contains keys less than 7: {1,2,3,4,5,6} - The right tree contains keys greater than or equal to 7: {7,8,9,10} **Final** **Left subtree (keys <7):** Rooted at 6 (splayed below 7 during rotation) **Right subtree (keys ≥7):** Root is 7 with right children 8, 9, 10 This completes the split operation at 7.