  We determine the limit of the expected value and the variance of the
protection number of the root in simply generated trees, in P\'olya trees, and
in unlabelled non-plane binary trees, when the number of vertices tends to
infinity. Moreover, we compute expectation and variance of the protection
number of a randomly chosen vertex in all those tree classes. We obtain exact
formulas as sum representations, where the obtained sums are rapidly converging
and therefore allowing an efficient numerical computation of high accuracy.
Most proofs are based on a singularity analysis of generating functions.
