On ideals related to Laver and Miller trees

In this work we consider the ideals $m^0(\mathcal{I})$ and ${\mathrm{\ell }}^0(\mathcal{I})$, ideals generated by the $\mathcal{I}$-positive Miller trees and $\mathcal{I}$-positive Laver trees, respectively. We investigate in which cases these ideals have cofinality larger than $\mathfrak{c}$ and we calculate some cardinal invariants closely related to these ideals.