Let C_n denote the number of nodes in a random binary search tree (of n nodes) at the maximal level. In this paper we present a direct proof of Robson's boundedness conjecture saying that the expected values EC_n remain bounded as n→∞. We also prove that EC_n is asymptotically (multiplicatively) periodic which shows that Robson's convergence conjecture (that is, EC_n is convergent) is only true if the limiting periodic function C˜(x) is constant. Interestingly, it can be shown that C˜(x) is almost constant in the sense that possible oscillations are very small. However, it seems to be a difficult problem to decide whether C˜(x) is really constant or not. We present similar properties for the variance of the height VarH_n, too.