  Using the construction by Bencs and T\'{o}th of invariant random subgroups on
weakly branch groups acting on regular rooted trees we produce uncountably many
indecomposable characters on these groups. In fact, we study three types of
characters coming from the action of a weakly branch group on a regular tree,
paying attention to their similarities and differences. We use obtained results
to show that each countable amenable branch group has uncountably many pairwise
not quasi-equivalent embeddings into Murray-von Neumann hyperfinite factor. For
the canonical character associated with a self-similar group and studied by the
second author as a self-similar trace we provide a number of examples when it
is explicitly computed.
