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A spanning tree of a graph is a k-additive tree spanner whenever the distance of every two vertices in the graph differs from that in the tree by at most k. In this paper we show that certain classes of graphs, such as distance-hereditary graphs, interval graphs, asteroidal triple-free graphs, allow some constant k such that every member in the class has some k-additive tree spanner. On the other hand, there are chordal graphs without a k-additive tree spanner for arbitrarily large k.