I'm reading Graph Polynomials by Shi et al. and this is included as a brief example of how evaluations of the interlace polynomials are applicable to families of graphs. The explanation provided is that since any one forest has exactly one perfect matching, and $q_N(G;1)$ for any simple graph $G$ gives the number of induced subgraphs of $G$ with an odd number of perfect matchings, that $q_N(F;1)$ must therefore count the total number of matchings for any forest $F$.
I'm not following the logic - would this not imply that $q_N(F;1)$ should be equal to the total number of induced subgraphs, since any induced subgraph of a forest is also a forest? I am also confused about the claim that every forest has one perfect matching - an easy counterexample is the path of length 2, which has no perfect matching. Am I misunderstanding the basic idea of a matching? Is there another way to think about this claim?