  We consider the quantity $P(G)$ associated with a graph $G$ that is defined
as the probability that a randomly chosen subtree of $G$ is spanning. Motivated
by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of
this graph invariant depending on the edge density, we establish first that
$P(G)$ is bounded below by a positive constant provided that the minimum degree
is bounded below by a linear function in the number of vertices. Thereafter,
the focus is shifted to the classical Erd\H{o}s-R\'enyi random graph model
$G(n,p)$. It is shown that $P(G)$ converges in probability to
$e^{-1/(ep_{\infty})}$ if $p \to p_{\infty} > 0$ and to $0$ if $p \to 0$.
