  Let $M$ be a $C$-minimal structure and $T$ its canonical tree (which
corresponds in an ultrametric space to the set of closed balls with radius
different than $\infty$ ordered by inclusion). We present a description of
definable locally constant functions $f:M\rightarrow T$ in $C$-minimal
structures having a canonical tree with infinitely many branches at each node
and densely ordered branches. This provides both a description of definable
subsets of $T$ in one variable and analogues of known results in algebraically
closed valued fields.
