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We continue the study of token sliding reconfiguration graphs of independentsets initiated by the authors in an earlier paper (arXiv:2203.16861). Two ofthe topics in that paper were to study which graphs $G$ are token slidinggraphs and which properties of a graph are inherited by a token sliding graph.In this paper we continue this study specializing on the case of when $G$and/or its token sliding graph $\mathsf{TS}_k(G)$ is a tree or forest, where$k$ is the size of the independent sets considered. We consider two problems.The first is to find necessary and sufficient conditions on $G$ for$\mathsf{TS}_k(G)$ to be a forest. The second is to find necessary andsufficient conditions for a tree or forest to be a token sliding graph. For thefirst problem we give a forbidden subgraph characterization for the cases of$k=2,3$. For the second problem we show that for every $k$-ary tree $T$ thereis a graph $G$ for which $\mathsf{TS}_{k+1}(G)$ is isomorphic to $T$. A numberof other results are given along with a join operation that aids in theconstruction of $\mathsf{TS}_k(G)$-graphs.

A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets