  Let $A$ be a commutative $k$-algebra over a field of $k$ and $\Xi$ a linear
operator defined on $A$. We define a family of $A$-valued invariants $\Psi$ for
finite rooted forests by a recurrent algorithm using the operator $\Xi$ and
show that the invariant $\Psi$ distinguishes rooted forests if (and only if) it
distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the
quantity $\alpha (T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the
generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with $U_n =\sum_{T\in
\bT_n} \frac 1{\alpha (T)} \Psi (T)$, where $\bT_n$ is the set of rooted trees
with $n$ vertices. We show that the generating function $U(q)$ satisfies the
equation $\Xi \exp U(q)= q^{-1} U(q)$. Consequently, we get a recurrent formula
for $U_n$ $(n\geq 1)$, namely, $U_1=\Xi(1)$ and $U_n =\Xi S_{n-1}(U_1, U_2,
>..., U_{n-1})$ for any $n\geq 2$, where $S_n(x_1, x_2, ...)$ $(n\in \bN)$ are
the elementary Schur polynomials. We also show that the (strict) order
polynomials and two well known quasi-symmetric function invariants of rooted
forests are in the family of invariants $\Psi$ and derive some consequences
about these well-known invariants from our general results on $\Psi$. Finally,
we generalize the invariant $\Psi$ to labeled planar forests and discuss its
certain relations with the Hopf algebra $\mathcal H_{P, R}^D$ in \cite{F}
spanned by labeled planar forests.
