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 . 4180538 6 n ≤ c ( n ) ≤ 1 . 4195988 1 n . Moreover, there is a strong indication that, in fact, a.a.s. c ( n ) ≤ 1 . 4180618 3 n .