Abstract
A ternary complex tree related to the golden ratio is used to show how the theory of complex trees works. We use the
topological set of this tree to obtain a parametric family of trees in one complex variable. Even though some real ferns
and leaves are reminiscent to elements of our family of study, here we only consider the underlying mathematics.
We provide esthetically appealing examples and a map of the unstable setM for this family. Moreover we show that
some elements found in the boundary of the unstable setM possess interesting algebraic properties, and we explain
how to compute the Hausdorff dimension and the shortest path of self-similar sets described by trees found outside
the interior of the unstable setM.
