Abstract
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.
Highlights
Mathematical trees have been studied for a very long time
We consider labelled t-ary trees, i.e. these are rooted trees embedded in the plane such that each vertex has a degree of at most t and the vertex set is [n] := {1, 2, . . . , n}
A vertex v is said to be reachable from a vertex i if there is an oriented path from vertex i to vertex v and we say that a path p has length if there are edges on the path
Summary
Mathematical trees (and their natural counterparts) have been studied for a very long time. From the proof of Proposition 1, it follows that there are 2 + d t(n + − 1) n+ −1 n− −d−1 unlabelled trees with a path of length starting at the root such that the terminating vertex has degree d. Summing over all j in Equation (9), we obtain the number of children of the root which are sinks of degree d, in trees of order n as (n − d − 2)!
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