  Given a positive integer $p$ and a graph $G$ with degree sequence
$d_1,\dots,d_n$, we define $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster
introduced a Tur\'an-type problem for $e_p(G)$: Given a positive integer $p$
and a graph $H$, determine the function $ex_p(n,H)$, which is the maximum value
of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$
as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the
classical Tur\'an number. Caro and Yuster determined the function $ex_p(n,
P_\ell)$ for sufficiently large $n$, where $p\geq 2$ and $P_\ell$ denotes the
path on $\ell$ vertices. In this paper, we generalise this result and determine
$ex_p(n,F)$ for sufficiently large $n$, where $p\geq 2$ and $F$ is a linear
forest. We also determine $ex_p(n,S)$, where $S$ is a star forest; and
$ex_p(n,B)$, where $B$ is a broom graph with diameter at most six.
