Trees as a tool for modelling undecidable problems

???????????? ?????????????? ? ??????? (????????????????) ?????? ??????? ???????? (??? ????????? ?????????? ??????? ?????? ??? ??????????? ? ????????? ????????, ??????? ?????????? ????? ??????) ? ????? ? ???????? ??????????? ??????, ??????????????? ?????, ??????????, ?????????? ????????????? ????????? ??????????????? ????? ?????? ??????, ??????? ???????????? ??? ????????? ?????????? ????????? ?????? ???? ?????????? ?????? ????. ????????, ??? ??? ????????? ?????????????? (??? ????????????????) ????????? ???????? ?????????? ????????? ???? ????????? ?????????, ???????????????? ?.?. ?????????? ?????? ????????? ????????????? ????????? ????????????; ????? ????????, ?????? ??????????? ??? ?????????. \medskip \begin{flushright} \textit{To Valentin Goranko, with gratitude from the author} \end{flushright}\medskip The paper proves undecidability and high (non-arithmeticity) of theories trees (with various refinements concept a tree requirements for properties trees, including finiteness number vertices) in language binary predicate letter corresponding to edges, equality, transitive closure operator congruence between pairs vertices, which is defined as equality distance vertices first pair second pair. It shown that many cases it sufficient apply only relation i.e., fact, instead operator, consider reachability relation; some cases, we establish without operator.