takes unitsthe access on the node with item will only take about / units instead of unitsthere are no nodes quite as deep as before an access on the node with item will bring nodes to within / of the rootand this is repeated until the depth becomes roughly log (an example with is too small to see the effect wellfigures to show the result of accessing items through in -node tree that originally contains only left children thus we do not get the same bad behavior from splay trees that is prevalent in the simple rotation strategy (actuallythis turns out to be very good case rather complicated proof shows that for this examplethe accesses take total of (ntime these figures highlight the fundamental and crucial property of splay trees when access paths are longthus leading to longer-than-normal search timethe rotations tend to be good for future operations when accesses are cheapthe rotations are not as good and can be bad the extreme case is the initial tree formed by the insertions all the insertions were constant-time operations leading to bad initial tree at that point in timewe had very bad treebut we were running ahead of schedule and had the compensation of less total running time then couple of really horrible accesses left nearly balanced treebut the cost was that we had to give back some of the time that had been saved the main theoremwhich we will prove in is that we never fall behind pace of (log nper operationwe are always on scheduleeven though there are occasionally bad operations figure result of splaying at node tree of all left children