Let ∈ (0, 1) and let G = (V G , E G ) be a graph. According to Dunbar et al. [ -Domination, Discrete Math. 211 (2000)
the minimum cardinality of an -dominating set of G and the -independent -domination number i (G) of G is the minimum cardinality of an -dominating set of G that is also -independent. A graph G is -domination perfect if (H ) = i (H ) for all induced subgraphs H of G.We characterize the -domination perfect trees in terms of their minimally forbidden induced subtrees. For ∈ (0, 1 2 ] there is exactly one such tree whereas for ∈ ( 1 2 , 1) there are infinitely many.