( that isn( ( )for any integer isuch that > since we already know the values of ( and ( )we pick so that is equal to either or that iswe pick by substituting the above value of in formula we obtainfor > ( by taking logarithms of both sides of formula we obtain log (hh/ from which we get logn( ( which implies that an avl tree storing entries has height at most logn by proposition and the analysis of binary search trees given in section the operations find and findallin dictionary implemented with an avl treerun in time (lognand (logn )respectivelywhere is the number of entries in the dictionary and is the size of the collection returned of coursewe still have to show how to maintain the height-balance property after an insertion or removal update operations