On algebraic sums, trees and ideals in the Cantor space

We work in the Cantor space 2^. The results of the paper adhere the following pattern. Let I \M, 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'₍--ₓ₈₌₄ₒ I for each n. 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.