  We consider a regular $n$-ary tree of height $h$, for which every vertex
except the root is labelled with an independent and identically distributed
continuous random variable. Taking motivation from a question in evolutionary
biology, we consider the number of simple paths from the root to a leaf along
vertices with increasing labels. We show that if $\alpha = n/h$ is fixed and
$\alpha > 1/e$, the probability there exists such a path converges to 1 as $h
\to \infty$. This complements a previously known result that the probability
converges to 0 if $\alpha \leq 1/e$.
