ix{series|({ ix{series!geometric|({ ix{euler' constant{ ix{series!geometric|){ ix{series!arithmetic|({ ix{series!arithmetic|){ ix{series!harmonic|({ ix{euler' constant{ ix{series!harmonic|){ ix{series|){ figure sample input for exercise euler' constant series - arithmetic - geometric harmonic figure sample output for exercise references more information on binary search treesand in particular the mathematical properties of treescan be found in the two books by knuth[ and [ several papers deal with the lack of balance caused by biased deletion algorithms in binary search trees hibbard' paper [ proposed the original deletion algorithm and established that one deletion preserves the randomness of the trees complete analysis has been performed only for trees with three nodes [ and four nodes [ eppinger' paper [ provided early empirical evidence of nonrandomnessand the papers by culberson and munro [ ][ provided some analytical evidence (but not complete proof for the general case of intermixed insertions and deletionsadelson-velskii and landis [ proposed avl trees recently it was shown that for avl treesif rebalancing is performed only on insertionsand not on deletionsunder certain circumstances the resulting structure still maintains depth of (log mwhere is the number of insertions [ simulation results for avl treesand variants in which the height imbalance is allowed to be at most for various values of kare presented in [ analysis of the average search cost in avl trees is incompletebut some results are contained in [ [ and [ considered self-adjusting trees like the type in section splay trees are described in [ -trees first appeared in [ the implementation described in the original paper allows data to be stored in internal nodes as well as leaves the data structure we have described