  The theory of complex trees is introduced as a new approach to study a broad
class of self-similar sets. Systems of equations encoded by complex trees
tip-to-tip equivalence relations are used to obtain one-parameter families of
connected self-similar sets $F_\mathcal{A}(z)$. In order to study topological
changes of $F_\mathcal{A}(z)$ in regions $\mathcal{R}\subset\mathbb{C}$ where
these families are defined, we introduce a new kind of set
$\mathcal{M}\subseteq\mathcal{R}$ which extends the usual notion of
connectivity locus for a parameter space. Moreover we consider another set
$\mathcal{M}_0\subseteq\mathcal{M}$ related to a special type of connectivity
for which we provide a theorem. Among other things, the present theory provides
a unified framework to families of self-similar sets traditionally studied as
separate with elements $F_\mathcal{A}(z)$ disconnected for parameters
$z\in\mathcal{R}\backslash\mathcal{M}$.
