  We consider an inhomogeneous oriented percolation model introduced by de
Lima, Rolla and Valesin. In this model, the underlying graph is an oriented
rooted tree in which each vertex points to each of its $d$ children with
`short' edges, and in addition, each vertex points to each of its $d^k$
descendant at a fixed distance $k$ with `long' edges. A bond percolation
process is then considered on this graph, with the prescription that
independently, short edges are open with probability $p$ and long edges are
open with probability $q$. We study the behavior of the critical curve
$q_c(p)$: we find the first two terms in the expansion of $q_c(p)$ as $k \to
\infty$, and prove that the critical curve lies strictly above the critical
curve of a related branching process, in the relevant parameter region. We also
prove limit theorems for the percolation cluster in the supercritical,
subcritical and critical regimes.
