The minimum number of inclusion-wise maximal distance-k independent sets in n-vertex trees is n if n ≤ k+1, and n - floor((n - (k mod 2)) / (floor(k/2) + 1)) + 1 otherwise.
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On the minimum number of maximal distance-$k$ independent sets in trees
The minimum number of inclusion-wise maximal distance-k independent sets in n-vertex trees is n if n ≤ k+1, and n - floor((n - (k mod 2)) / (floor(k/2) + 1)) + 1 otherwise.