In "The Potts model and the Tutte Polynomial", D.J.A. Welsh and C. Merino claim on pg. 1135, equation 18, that, \begin{align*} \sum_{i=0}^{n-1}f_i t^i = t^{n-1}T_G(1+\frac{1}{t},1) \end{align*} where $T_G(x,y)$ is the Tutte polynomial of some graph $G$ and $f_i$ denotes the number of forests of $G$ with exactly $i$ edges. Their paper is available online here.

Welsh and Merino do not give any citations or proof for this claim. Does anyone know of such a proof? Also, is there a more explicit formula for $f_i$ in terms of $T_G(x,y)$?

EDIT: I noticed recently that something must be wrong with the claimed formula because if $G$ is the disjoint union of $n$ isolated vertices, then $T_G(x,y) = 1$, so the righthand side becomes $t^{n-1}$; however the lefthand side is $1$. Implicitly Welsh and Merino are assuming that $G$ is connected or I do not know what is meant by the "size" of a forest.