  Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an
$F$-covering if for every vertex in $G$ there is a copy of $F$ covering it. For
$1\leq i\leq k-1$, the minimum $i$-degree $\delta_i(G)$ of $G$ is the minimum
integer such that every $i$ vertices are contained in at least $\delta_i(G)$
edges. Let $c_i(n,F)$ be the largest minimum $i$-degree among all $n$-vertex
$k$-uniform hypergraphs that have no $F$-covering. In this paper, we consider
the $F$-covering problem in $3$-uniform hypergraphs when $F$ is the generalized
triangle $T$, where $T$ is a $3$-uniform hypergraph with the vertex set
$\{v_1,v_2,v_3,v_4,v_5\}$ and the edge set
$\{\{v_{1}v_{2}v_{3}\},\{v_{1}v_{2}v_{4}\},\{v_{3}v_{4}v_{5}\}\}$. We give the
exact value of $c_2(n,T)$ and asymptotically determine $c_1(n,T)$. We also
consider the $F$-covering problem in $3$-uniform hypergraphs when $F$ are some
trees, such as the linear $k$-path $P_k$ and the star $S_k$. Especially, we
provide bounds of $c_i(n,P_k)$ and $c_i(n,S_k)$ for $k\geq 3$, where $i=1,2$.

Combinatorics