  In this paper we study a variation of the accessibility percolation model,
this is also motivated by evolutionary biology and evolutionary computation.
Consider a tree whose vertices are labeled with random numbers. We study the
probability of having a monotone subsequence of a path from the root to a leaf,
where any $k$ consecutive vertices in the path contain at least one vertex of
the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at
distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary
trees, we prove that, as $h$ tends to infinity the probability of having such
subsequence: tends to 1, if $n$ grows significantly faster than
$\sqrt[k]{h/(ek)}$; and tends to 0, if $n$ grows significantly slower than
$\sqrt[k]{h/(ek)}$.
