In this study two actual types of problems are considered and solved: 1) determining the maximum common connected fragment of the T-tree (T-directed tree) which does not change with time; 2) determining all non-isomorphic maximum common connected fragments of the T-tree (T-directed tree) which do not change with time. The choice of the primary study of temporal directed trees and trees is justified by the wide range of their practical applications. Effective methods for their solution are proposed. Examples of the solution of the problem for temporal trees and temporal directed trees are given. It is shown that the experimental estimates of the computational complexity of the solution for problems of the temporal directed trees and the temporal trees.