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We show that every polynomial in $\mathbb Z[x]$ defines an endomorphism ofthe $d$-ary rooted tree induced by its action on the ring $\mathbb Z_d$ of$d$-adic integers. The sections of this endomorphism also turn out to beinduced by polynomials in $\mathbb Z[x]$ of the same degree. In the case ofpermutational polynomials acting on $\mathbb Z_d$ by bijections the inducedendomorphisms are automorphisms of the tree. In the case of $\mathbb Z_2$ suchpolynomials were completely characterized by Rivest. As our main application weutilize the result of Rivest to derive the condition on the coefficients of apermutational polynomial $f(x)\in\mathbb Z[x]$ that is necessary and sufficientfor $f$ to induce a level transitive automorphism of the binary tree, which isequivalent to the ergodicity of the action of $f(x)$ on $\mathbb Z_2$ withrespect to the normalized Haar measure.

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers