  The necessary and sufficient conditions under which a given family
$\mathcal{F}$ of subsets of finite set $X$ coincides with the family
$\mathbf{B}_X$ of all balls generated by some ultrametric $d$ on $X$ are found.
It is shown that the representing tree of the ultrametric space
$(\mathbf{B}_{X}, d_H)$ with the Hausdorff distance $d_H$ can be obtained from
the representing tree $T_X$ of ultrametric space $(X, d)$ by adding a leaf to
every internal vertex of $T_X$.
