The maximum number of maximum generalized 4-independent sets in trees

A generalized k-independent set is a set of vertices such that the induced subgraph contains no trees with k-vertices, and the generalized k-independence number is the cardinality of a maximum k-independent set in G. Zito proved that the maximum number of maximum generalized 2-independent sets in a tree of order n is 2(n-3/2) if n is odd, and 2(n-2/2) +1 if n is even. Tu et al. showed that the maximum number of maximum generalized 3-independent sets in a tree of order n is 3(n/3-1)+n/3+1 if n equivalent to 0 (mod 3), and 3(n-1/3)-1+1 if n equivalent to 1 (mod 3), and 3(n-2/3)-1 if n equivalent to 2 (mod 3) and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized k-independent sets in a tree for a general integer k. As applications, we show that the maximum number of generalized 4-independent sets in a tree of order n(n >= 4) is {4(n-4/4)+(n/4+1)(n/8+1), n equivalent to 0 (mod4), 4(n-5/4)+n-1/4+1, n equivalent to 1 (mod 4), 4(n-6/4)+n-2/4, n equivalent to 2 mod 4), 4(n-7/4), n equivalent to 3 (mod 4) and we also characterize the structure of all extremal trees with the maximum number of maximum generalized 4-independent sets.