  Inspired by natural classes of examples, we define generalized directed
semi-tree and construct weighted shifts on the generalized directed semi-trees.
Given an $n$-tuple of directed directed semi-trees with certain properties, we
associate an $n$-tuple of multiplication operators on a Hilbert space
$\mathscr{H}^2(\beta)$ of formal power series.
  Under certain conditions, $\mathscr{H}^2(\beta)$ turns out to be a
reproducing kernel Hilbert space consisting of holomorphic functions on some
domain in $\mathbb C^n$ and the $n$-tuple of multiplication operators on
$\mathscr{H}^2(\beta)$ is unitarily equivalent to an $n$-tuple of weighted
shifts on the generalized directed semi-trees. Finally, we exhibit two classes
of examples of $n$-tuple of operators which can be intrinsically identified as
weighted shifts on generalized directed semi-trees.
