Metrics transformations preserving the types of one-dimensional minimal fillings

Given a class F of metric spaces and a family of transformations T of a metric, one has to describe a family of transformations T' ? T that transfer F into itself and preserve some types of minimal fillings. The article considers four cases. First, when F is the class of all finite metric spaces, T = {(M,?) ? (M, f??) | f : R>0 ? R>0}, and the elements of T' preserve all non-degenerate types of minimal fillings of four-point metric spaces and finite non-degenerate stars, and we prove that T' = {(M,?) ? (M,?? + a):a > ?a?}. Second, when F is the class of all finite metric spaces, the class T consists of the maps ? ? N?, where the matrix N is the sum of a positive diagonal matrix A and a matrix with the same rows of non-negative elements. The elements of T' preserve all minimal fillings of the type of non-degenerate stars. It has been proven that T0 consists of maps ? ? N? where A is scalar. Third, when F is the class of all finite additive metric spaces, T is the class of all linear mappings given by matrices, and the elements of T' preserve all non-degenerate types of minimal fillings, and we proved that for metric spaces consisting of at least four points T' is the set of transformations given by scalar matrices. Fourth, when F is the class of all finite ultrametric spaces, T is the class of all linear mappings given by matrices, and we proved that for threepoint spaces the matrices have the form A = R(B + ?E), where B is a matrix of identical rows of positive elements, and R is a permutation of the points (1,0,0),(0,1,0) and (0,0,1).