  Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c =
(S_i)_{i\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\ldots, S_{k}$. Given a
weight function $w: V(G) \to \mathbb{R}^+$, the weight of a color $S_i$ is
defined as $w(i) = \max_{v \in S_i} w(v)$ and the weight of a coloring $c$ as
$w(c) = \sum_{i=1}^{k}w(i)$. Guan and Zhu [Inf. Process. Lett., 1997] defined
the weighted chromatic number of a pair $(G,w)$, denoted by $\sigma(G,w)$, as
the minimum weight of a proper coloring of $G$. For a positive integer $r$,
they also defined $\sigma(G,w;r)$ as the minimum of $w(c)$ among all proper
$r$-colorings $c$ of $G$.
  The complexity of determining $\sigma(G,w)$ when $G$ is a tree was open for
almost 20 years, until Ara\'ujo et al. [SIAM J. Discrete Math., 2014] recently
proved that the problem cannot be solved in time $n^{o(\log n)}$ on $n$-vertex
trees unless the Exponential Time Hypothesis (ETH) fails.
  The objective of this article is to provide hardness results for computing
$\sigma(G,w)$ and $\sigma(G,w;r)$ when $G$ is a tree or a forest, relying on
complexity assumptions weaker than the ETH. Namely, we study the problem from
the viewpoint of parameterized complexity, and we assume the weaker hypothesis
$FPT \neq W[1]$. Building on the techniques of Ara\'ujo et al., we prove that
when $G$ is a forest, computing $\sigma(G,w)$ is $W[1]$-hard parameterized by
the size of a largest connected component of $G$, and that computing
$\sigma(G,w;r)$ is $W[2]$-hard parameterized by $r$. Our results rule out the
existence of $FPT$ algorithms for computing these invariants on trees or
forests for many natural choices of the parameter.
