The "forest" in her name is an abbreviation of "雑木林", a nickname of the Matsu-class destroyers. This is difficult to translate but refers to a diverse forest with many different species of trees, or a "mixed" forest. The Matsu-class were so-called because each was named after a different tree species, and there were many of them (74 were planned and 18 completed, without even counting the Tachibana-class as a subclass).
