Sub-trees of a random tree

Let T be a random tree taken uniformly at random from the family of labelled trees on n vertices. In this note, we provide bounds for c(n), the number of sub-trees of T that hold asymptotically almost surely (a.a.s.). With computer support we show that a.a.s. 1.41805386n≤c(n)≤1.41959881n. Moreover, there is a strong indication that, in fact, a.a.s. c(n)≤1.41806183n.