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.