  The expansion of a graph $F$, denoted by $F^3$, is the $3$-graph obtained
from $F$ by adding a new vertex to each edge such that different edges receive
different vertices. For large $n$, we establish tight upper bounds for:
  The maximum number of edges in an $n$-vertex $3$-graph that does not contain
$T^3$ for certain class $\mathcal{T}$ of trees, sharpening (partially) a result
of Kostochka--Mubayi--Verstra\"{e}te.
  The minimum number of colors needed to color the complete $n$-vertex
$3$-graph to ensure the existence of a rainbow copy of $F^3$ when $F$ is a
graph obtained from some tree $T\in \mathcal{T}$ by adding a new edge,
extending anti-Ramsey results on $P_{2t}^3$ by Gu--Li--Shi and $C_{2t}^3$ by
Tang--Li--Yan.
  The maximum number of edges in an $n$-vertex $3$-graph whose shadow does not
contain the shadow of $C_{k}^3$ or $T^3$ for $T\in \mathcal{T}$, answering a
question of Lv \etal on generalized Tur\'{a}n problems.
