allow ${\ensuremath{\mathcal S}}$ to contain rotations at a finger in $T$ (although $k$-finger strategies as described above do not generate rotations). The position of a finger is maintained during a rotation.

We show how ${\ensuremath{\mathcal S}}$ can be simulated in a standard dynamic BST. If in ${\ensuremath{\mathcal S}}$ a finger visits a node, then the (single) pointer in the BST also visits the corresponding node, therefore all accesses are correctly served in the BST. Every elementary step in ${\ensuremath{\mathcal S}}$ is mapped to (amortized) $O(\log{k})$ elementary steps (pointer moves and rotations) in the BST. This immediately implies Theorem \[thm: LF simulation\], since, if we can simulate an arbitrary $k$-finger execution, then indeed we can simulate the optimal $k$-finger execution on the best static tree. Assuming that the intial conditions $T$ and $\vec{\ell}$ are known, the steps of ${\ensuremath{\mathcal S}}$ are simulated one-by-one, without any lookahead. Thus, insofar as the $k$-finger execution is *online*, the BST execution is also online (this fact is used in §\[sec3\]).

Let us describe simulation by a standard BST $T'$ of a $k$-finger execution ${\ensuremath{\mathcal S}}$ in a BST $T$. The construction is a refinement of the one given by Demaine et al. [@CombineBST], see also 