Random subtrees and unimodal sequences in graphs

For a complete graph $K_n$ and a nonnegative integer k, we study the probability that a random subtree of $K_n$ has exactly $n-k$ vertices and show that it approaches a limiting value of $\frac{e^{-k-e^{-1}}}{k!}$ as n tends to infinity. We also consider the (conditional) probability that a random subtree of $K_n$ contains a given edge, and more generally, a fixed subtree. In particular, if e and f are adjacent edges of $K_n$, Chin, Gordon, MacPhee and Vincent [J. Graph Theory 89 (2018), 413-438] conjectured that $P[e\subseteq T|f\subseteq T]\le P[e\subseteq T]$. We prove this conjecture and further prove that $P[e\subseteq T|f\subseteq T]$ tends to three-quarters of $P[e\subseteq T]$ as $n\to \infty$. Finally, several classes of graphs are given, such as star plus an edge, lollipop graph and glasses graph, whose subtree polynomials are unimodal.