Ultrametric preserving functions and weak similarities of ultrametric spaces

Let $WS(X, d)$ be the class of ultrametric spaces which are weakly similar to ultrametric space $(X, d)$. The main results of the paper completely describe the ultrametric spaces $(X, d)$ for which the equality $$\rho(x, y) = f(d(\Phi(x), \Phi(y)))$$ holds for every $(Y, \rho) \in WS(X, d)$, every weak similarity $\Phi \colon Y \to X$, and all $x$, $y \in Y$ with some ultrametric (pseudoultrametric) preserving function $f$ depending on $\Phi$.