Reachability results in labelled \(t\)-ary trees
Isaac Owino Okoth, Albert Oloo Nyariaro · Open Journal of Mathematical Sciences · 2021
In this paper, we prove some new formulas in the enumeration of labelled \(t\)-ary trees by path lengths. We treat trees having their edges oriented from a vertex of lower label towards a vertex of higher label. Among other results, we obtain counting formulas for the number of \(t\)-ary trees on \(n\) vertices in which there are paths of length \(\ell\) starting at a root with label \(i\) and ending at a vertex, sink, leaf sink, first child, non-first child and non-leaf. For each statistic, the average number of these reachable vertices is obtained for any random \(t\)-ary tree.