Maximum Distance in Random Search

(a) For the cases n = 1, 2, 3 we observe that ∆n = 0, 1, 2, respectively, irrespective of the structure of the random search tree. We conclude that δ1 = 0, δ2 = 1, δ3 = 2. For the case n = 4 we observe that either ∆4 = 2 or ∆4 = 3, depending on the structure of the random search tree (∆4 > 2 because we already have ∆3 > 2 and the longest distance cannot decrease by adding more nodes, ∆4 6 3 because there are only three edges in total). The only search trees with ∆4 = 2 are and 1 . Hence, with the help of Lemma 1.1,