  We study certain phase transitions of branching random walks (BRW) on Cayley
graphs of free products. The aim of this paper is to compare the size and
structural properties of the trace, i.e., the subgraph that consists of all
edges and vertices that were visited by some particle, with those of the
original Cayley graph. We investigate the phase when the growth parameter
$\lambda$ is small enough such that the process survives but the trace is not
the original graph. A first result is that the box-counting dimension of the
boundary of the trace exists, is almost surely constant and equals the
Hausdorff dimension which we denote by $\Phi(\lambda)$. The main result states
that the function $\Phi(\lambda)$ has only one point of discontinuity which is
at $\lambda_{c}=R$ where $R$ is the radius of convergence of the Green function
of the underlying random walk. Furthermore, $\Phi(R)$ is bounded by one half
the Hausdorff dimension of the boundary of the original Cayley graph and the
behaviour of $\Phi(R)-\Phi(\lambda)$ as $\lambda \uparrow R$ is classified.
  In the case of free products of infinite groups the end-boundary can be
decomposed into words of finite and words of infinite length. We prove the
existence of a phase transition such that if $\lambda\leq \tilde\lambda_{c}$
the end boundary of the trace consists only of infinite words and if
$\lambda>\tilde\lambda_{c}$ it also contains finite words. In the last case,
the Hausdorff dimension of the set of ends (of the trace and the original
graph) induced by finite words is strictly smaller than the one of the ends
induced by infinite words.
