We continue the investigation of the Laver ideal ℓ 0 and Miller ideal m 0 started in [GJSp] and [GRShSp]; these are the ideals on the Baire space associated with Laver forcing and Miller forcing. We solve several open problems from these papers. The main result is the construction of models for $t , where add denotes the additivity coefficient of an ideal. For this we construct amoeba forcings for these forcings which do not add Cohen reals. We show that (...) c = ω 2 implies $\operatorname{add}(m^0) \leq \mathfrak{h}$ . We show that b = c, d = c implies $\operatorname{cov}(\ell^0) \leq \mathfrak{h}^+, \operatorname{cov}(m^0) \leq \mathfrak{h}^+$ respectively. Here $\operatorname{cov}$ denotes the covering coefficient. We also show that in the Cohen model $\operatorname{cov}(m^0) holds. Finally we prove that Cohen forcing does not add a superperfect tree of Cohen reals. (shrink)