On Generalized Independent Subsets of Trees

A natural generalization of the widely discussed independent (or “internally stable”) subsets of graphs is to consider subsets of vertices where no two elements have distance less or equal to a fixed number k (“k-independent subsets”). In this paper we give asymptotic results on the average number of ˆ-independent subsets for trees of size n, where the trees are taken from a so-called simply generated family. This covers a lot of interesting examples like binary trees, general planted plane trees, and others.