The center of distances of a metric space (𝑋,𝑑) is the set 𝐶(𝑋) of all 𝑡∈ℝ+ for which the equation 𝑑(𝑥,𝑝)=𝑡 has a solution for each 𝑝∈𝑋. We prove that the equalities 𝐶(𝑋)={0} or 𝐶(𝑋)={0,diam𝑋} hold if (𝑋,𝑑) is an ultrametric space generated by labeled trees. The necessary and sufficient conditions under which diam𝑋∈𝐶(𝑋) are found.