Definability in the -quasiorder of labeled forests

We prove that for any k ≥ 3 each element of the h -quasiorder of finite k -labeled forests is definable in the ordinary first order language and, respectively, each element of the h -quasiorder of (at most) countable k -labeled forests is definable in the language L ω 1 ω , in both cases provided that the minimal non-smallest elements are allowed as parameters. As corollaries, we characterize the automorphism groups of both structures and show that the structure of finite k -forests is atomic. Similar results hold true for two other relevant structures: the h -quasiorder of finite (resp. countable) k -labeled trees and of finite (resp. countable) k -labeled trees with a fixed label of the root element.