  In this paper, we consider covering graphs obtained by lifting a tree with a
loop at each vertex as a voltage graph over a cyclic group. We generalize a
tool of Hell, Nishiyama, and Stacho, known as the billiard strategy, for
constructing Hamiltonian cycles in the covering graphs of paths. We show that
our extended tool can be used to provide new sufficient conditions for the
Hamiltonicity of covering graphs of trees that are similar to those of Batagelj
and Pisanski and of Hell, Nishiyama, and Stacho. Next, we focus specifically on
covering graphs obtained from trees lifted as voltage graphs over cyclic groups
$\mathbb Z_p$ of large prime order $p$. We prove that for a given reflexive
tree $T$ whose edge labels are assigned uniformly at random from a finite set,
the corresponding lift is almost surely Hamiltonian for a large enough
prime-ordered cyclic group $\mathbb Z_p$. Finally, we show that if a reflexive
tree $T$ is lifted over a group $\mathbb Z_p$ of a large prime order, then for
any assignment of nonzero elements of $\mathbb Z_p$ to the edges of $T$, the
corresponding cover of $T$ has a large circumference.
