Turán number for bushes

Let a, b ∈ Z⁺, r = a+b, and let T be a tree with color classes U = {u₁, u₂, …, u_s} and V = {v₁, v₂, …, v_t}. Let A₁, …, A_s and B₁, …, B_t be disjoint sets, such that |A_i | = a and |B_j | = b for all i, j. The (a, b)-blowup of T is the r-uniform hypergraph with edge set {A_i ∪ B_j: u_iv_j ∈ E(T)}. We use the ∆-systems method to prove the following Turán-type result. Suppose a, b, t ∈ Z⁺, r = a + b ≥ 3, a ≥ 2, and T is a fixed tree of diameter 4 in which the degree of the center vertex is t. Then there exists a C = C(r, t, T) > 0 such that |E(H)| ≤ (t − 1)) n r−1) + Cn^r−2 for every n-vertex r-uniform hypergraph H not containing an (a, b)-blowup of T. This is asymptotically exact when t ≤ |V (T)|/2. A stability result is also presented.