Reachability Results in Plane Trees

In this paper, we obtain closed formulas for the number of reachable vertices in labelled plane trees by paths lengths, sinks, leaf first children, left most path, non-first and non-leaves. Our counting objects are having their edges oriented from a vertex lower label towards higher label. For each statistic, average vertices. Moreover, formula on $n$ such that exactly $k\leq n$ root.