  A linear $r$-uniform hypergraph is called acycilc if it can be constructed
starting from one single edge then at each step adding a new edge that
intersect the union of the vertices of the previous edges in at most one
vertex. Recently, Gy\'{a}rf\'{a}s, Ruszink\'{o} and S\'{a}rk\''{o}zy initiated
the study of the linear Tur\'{a}n numbers of acyclic linear triple systems. In
this paper, we extend their results to linear quadruple systems. Here, we
concentrate on small trees, paths and matchings. For the case of small trees,
we find that for a linear tree $T$, $ex^{lin}_{4}(n,T)$ relates to difficult
problems on Steiner system $S(2,4,n)$ For example, we show that
$ex^{lin}_{4}(n, P_4)\le \frac{5n}{4}$ with equality holds if and only if the
linear quadruple system is the disjoint union of $S(2,4,16)$. Denote by
$E^{+}_{4}$ the linear tree consisting of three pairwise disjoint quadruples
and a fourth one intersecting all of them. We prove that
$12\lfloor\frac{n-4}{9}\rfloor\le ex^{lin}_{4}(n, E^{+}_4)\le
\frac{14(n-s)}{9}$, where $s$ is the number of vertices in $G$ with degree at
least 8. Denote by $M_k$ and $P_k$ the set of $k$ pairwise disjoint quadruples
and the linear path with $k$ quadruples, respectively. For the case of paths,
we show that $ex^{lin}_{4}(n, P_k)\le 2.5kn$. For the case of matchings, we
prove that for fixed $k$ and sufficiently large $n$, $ex^{lin}_{4}(n,
M_k)=g(n,k)$ where $g(n,k$) denotes the maximum number of quadruples that can
intersect $k-1$ vertices in a linear quadruple system on $n$ vertices.
