We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

We use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given in [5].

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 ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → 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.