For a group $G$ acting on a regular rooted $d$-ary tree $T_d$ and on its boundary $\partial T_d$ we consider the diagonal actions of $G$ on the powers of $T_d$ and $\partial T_d$. For the action of the full group $\mathrm{Aut}(T_d)$ of automorphisms of $T_d$ we describe the ergodic decomposition of its action on $(\partial T_d)^n$ for all $n\geq 1$. To achieve it we analyze the orbits of $n$-tuples of elements of vertices of any fixed finite level of $T_d$. For a subgroup $G$ of $\mathrm{Aut}(T_d)$ the corresponding orbits may be smaller, but sometimes they coincide with the orbits of the full group of automorphisms for all levels. In this case we say that the action of $G$ on $\mathrm{Aut}(T_d)$ is maximally tree $n$-transitive. For example, maximal tree 1-transitivity is equivalent to level transitivity of the action of $G$ on $T_d$. It follows from the results of Bartholdi and Grigorchuk that Grigorchuk group and Basilica group act maximally tree 2-transitively on $\partial T_2$. We show that the action of Grigorchuk group on $\partial T_2$ is, in-fact, maximally tree 4-transitive but not maximally 5-transitive. The talk is based on a joint work with Rostislav Grigorchuk and Zoran \v Suni\’c.