Let H be an r-uniform hypergraph and F be a graph. We say H contains F as a trace if there exists some set S ⊆ V (H) such that H|S := {E ∩ S : E ∈ E(H)} contains a subgraph isomorphic to F. Let exr(n, T r(F )) denote the maximum number of edges of an n-vertex r-uniform hypergraph H which does not contain F as a trace. In this paper, we improve the lower bounds of exr(n, T r(F )) when F is a star, and give some optimal cases. We also improve the upper bound for the case when H is 3-uniform and F is K2,t when t is small.