Counting Special Families of Labelled Trees

Abstract

A labelled tree rooted at its least labelled vertex is Least-Child-Being-Monk if it has the property that the least labelled child of 0 is a leaf. One of our main results is that the number of Least-Child-Being-Monk trees labelled on {0, 1, 2,... ,n + 1} is equal to n n . More generally, let $$ \mathcal{T}_{{n + 1,p}} $$ be the set of labelled trees on {0,1,2,..., n + 1}, such that the total number of descendants of the least labelled child of 0 is p. We prove that the cardinality of $$ \mathcal{T}_{{n + 1,p}} $$ is equal to $$ (n - p)^{{n - p}} (p + 1)^{{p - 1}} {\left( {\begin{array}{*{20}c} {{n + 1}} \\ {p} \\ \end{array} } \right)}. $$ Furthermore, a labelled tree rooted at its least labelled vertex is Hereditarily-Least-Single if it has the property that every least child in this tree is a leaf. Let the number of Hereditarily-Least-Single trees with n vertices be h n . We find a functional equation for the generating function of h(n) and derive a recurrence that will quickly compute h(n).