Definability of closure operations in the $h$-quasiorder of labeled forests

We prove that natural closure operations on quotient structures of the $h$-quasiorder of finite and (at most) countable $k$-labeled forests ($k\ge3$) are definable provided that minimal nonsmallest elements are allowed as parameters. This strengthens our previous result which holds that each element of the $h$-quasiorder of finite $k$-labeled forests is definable in the first-order language, and each element of the $h$-quasiorder of (at most) countable $k$-labeled forests is definable in the language $L_{\omega_1\omega}$; in both cases $k\ge3$ and minimal nonsmallest elements are allowed as parameters. Similar results hold true for two other relevant structures: the $h$-quasiorder of finite (resp. countable) $k$-labeled trees and $k$-labeled trees with a fixed label on the root element.