sentences:  Our upper bound will be formula_4, where "k" is the number of interleaves.The total cost is divided into two parts, searching for the element, and updating the structure of the tango tree to maintain the proper invariants (switching preferred children and re-arranging preferred paths).To see that the searching (not updating) fits in this bound, simply note that every time an auxiliary tree search is unsuccessful and we have to move to the next auxiliary tree, that results in a preferred child switch (since the parent preferred path now switches directions to join the child preferred path).