  An equitable tree-$k$-coloring of a graph is a vertex coloring on $k$ colors
so that every color class incudes a forest and the sizes of any two color
classes differ by at most one.This kind of coloring was first introduced in
2013 and can be used to formulate the structure decomposition problem on the
communication network with some security considerations. In 2015, Esperet,
Lemoine and Maffray showed that every $d$-degenerate graph admits an equitable
tree-$k$-coloring for every $k\geq 3^{d-1}$. Motivated by this result, we
attempt to lower their exponential bound to a linear bound. Precisely, we prove
that every $d$-degenerate graph $G$ admits an equitable tree-$k$-coloring for
every $k\geq \alpha d$ provided that $|G|\geq \beta \Delta(G)$, where
$(\alpha,\beta)\in \{(8,56), (9,26), (10,18), (11,15), (12,13), (13,12),
(14,11), (15,10), (17,9), (20,8), (27,7), (52,6)\}$.
