On algebraic sums, trees and ideals in the Baire space

We work in the Baire space ${\mathbb{Z}}^{\omega }$ equipped with the coordinate-wise addition $+$. Consider a $\sigma -$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in \mathbb{T}$ there exists $T^{\mathrm{\prime }}\in \mathbb{T},T^{\mathrm{\prime }}\subseteq T$ such that $A+\underset{\text{n–times}}{\underbrace{[T^{\mathrm{\prime }}]+[T^{\mathrm{\prime }}]+\cdots +[T^{\mathrm{\prime }}]}}\in \mathcal{I}$ for each $n\in \omega$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $\omega -$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $\sigma$-ideal $\mathcal{M}$ of meager sets, we also analyze ${\mathcal{M}}_{-}$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.