  We consider the eternal distance-2 domination problem, recently proposed by
Cox, Meger, and Messinger, on trees. We show that finding a minimum eternal
distance-2 dominating set of a tree is linear time in the order of the graph by
providing a fast algorithm. Additionally, we characterise when trees have an
eternal distance-2 domination number equal to their domination number or their
distance-2 domination number, along with characterizing which trees are eternal
distance-2 domination critical. We conclude by providing general upper and
lower bounds for the eternal distance-k domination number of a graph, as well
as constructing an infinite family of trees which meet said upper bound and
another which meets the given lower bound.
