Splay tree rotation algorithm: Why use zig-zig and zig-zag instead of simpler rotations?

The algorithm you're proposing on the tree is called the "move-to-root" heuristic and is discussed on page four of Sleator and Tarjan's original paper on splay trees. They cite an older paper by Allen and Munro where it is shown that if you try to use move-to-root as a means for reshaping trees, it is possible for the amortized cost of each lookup to be O(n), which is quite slow. Splaying is a very carefully designed algorithm for reshaping the tree, and it guarantees amortized O(log n) lookups no matter what sequence of accesses is performed.

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