How many random questions are necessary to identify \(n\) distinct objects? (Q1813292)

Let \(H_ n^{(1)}\) and \(H_ n^{(2)}\) be respectively the random number of necessary questions for two different models needed to determine the unknown bijective mapping \(\phi: X\to A\). In this paper it has been proved that \[ H_ n^{(1)}=2\log_ 2 n+O_ e(1) \] and \[ H_ n^{(2)}=\log_ 2 n+[1+O_ e(1)](2\log_ 2 n)^{1/2}, \] where \(O_ e(1)\to\infty\) as \(n\to\infty\). A connection between the above models and digital search tree in computer science has been established.