How rigid the finite ultrametric spaces can be?

A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $\mathrm{\mid }X\mathrm{\mid }\ge 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $\mathrm{\mid }X\mathrm{\mid }-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $\mathrm{\mid }X\mathrm{\mid }-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.