On algebraic sums, trees and ideals in the Cantor space

We work in the Cantor space 2^ω. The results of the paper adhere the following pattern. Let I∈ , N, M∩ N, E\ and T be a perfect, uniformly perfect or Silver tree. Then for every A∈ I there exists T'⊆ T of the same kind as T such that A+[T']+[T']+ +[T']ₙ₋₋ₜᵢₘₑₛ∈ I for each n∈ω. We also prove weaker statements for splitting trees. For the case E we also provide a simple characterization of basis of E. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpi\'nski set belongs to u₀ and v₀, provided that c is a regular cardinal.