  In this paper we consider a notion of nonmeasurablity with respect to
Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, and $cl_0$. We
show that there exists a subset $A$ of the Baire space $\omega^\omega$ which is
$s$-, $l$-, and $m$-nonmeasurable, that forms dominating m.e.d. family. We
introduce and investigate a notion of $\mathbb{T}$-Bernstein sets - sets that
intersect but does not containt any body of a tree from a given family of trees
$\mathbb{T}$. We also acquire some results on $\mathcal{I}$-Luzin sets, namely
we prove that there are no $m_0$-, $l_0$-, and $cl_0$-Luzin sets and that if
$\mathfrak{c}$ is a regular cardinal, then the algebraic sum (considered on the
real line $\mathbb{R}$) of a generalized Luzin set and a generalized
Sierpi\'nski set belongs to $s_0, m_0$, $l_0$ and $cl_0$.
