Vector optimization of decompositions of root trees

We consider the decompositions of root trees with vector vertex weights. We study a problem on the minimization of the number of the decomposition under a vector constraint. We prove the insolvability of this problem in a class of generalized finite automata over trees containing algorithms of gradient type. We estimate the error of the automaton algorithm, present an algorithm that solves a problem with polynomially bounded time complexity, and obtain an estimate for the number of parts of the decomposition.