  We classify $\mathcal{R}$- and $\mathcal{L}$-cross-sections of partial wreath
product of inverse semigroups. As a corollary, we get the description of
$\mathcal{R}$- and $\mathcal{L}$-cross-sections of the semigroupof partial
automorphisms of finite regular rooted tree and compute also the number of
different $\mathcal{R}$- ($\mathcal{L}$-) cross-sections in this semigroup.
