In their paper Asymmetric Graphs Erdős and Rényi proved that almost all trees have non-trivial automorphism group. More specifically they showed that almost all trees contain at least one so-called cherry, i.e. three vertices $i, j, k$ where $i, j$ are leaves, $k$ has degree $3$ and $(i,k)$ and $(j,k)$ are edges.
Unfortunately I'm having some trouble understanding their proof. I can follow the counting argument but fail to understand the following bits, in particular the part where Chebyshev's inequality is used. I would be very grateful if someone could explain to me the steps in the proof in more detail or point me to some other reference for this result. I haven't found this in any graph theory book.
I have linked the original paper above. The relevant bit starts on page 18 of the pdf.