The asymptotic number of non-isomorphic rooted trees obtained by rooting a tree

Let Tn denote the set of trees with n vertices. Suppose that each tree in Tn is equally likely. We show that the number of non-isomorphic rooted trees obtained by rooting a tree equals (μr+o(1))n for almost every tree of Tn, where μr is a constant. As an application, we show that in Tn the number of any given pattern, which is a fixed small tree with internal vertices specified, is asymptotically normally distributed with mean ∼μMn and variance ∼σMn, where μM and σM are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.