  Using event-driven kinetic Monte-Carlo simulations we investigate the early
stage of non-equilibrium surface growth in a generic model with anisotropic
interactions among the adsorbed particles. Specifically, we consider a
two-dimensional lattice model of spherical particles where the interaction
anisotropy is characterized by a control parameter $\eta$ measuring the ratio
of interaction energy along the two lattice directions. The simplicity of the
model allows us to study systematically the effect and interplay between
$\eta$, the nearest-neighbor interaction energy $E_{n}$, and the flux rate $F$,
on the shapes and the fractal dimension $D_{f}$ of clusters before coalescence.
At finite particle flux $F$ we observe the emergence of rod-like and
needle-shaped clusters whose aspect ratio $R$ depends on $\eta$, $E_{n}$ and
$F$. In the regime of strong interaction anisotropy, the cluster aspect ratio
shows power-law scaling as function of particle flux, $R \sim F^{- \alpha}$.
Furthermore, the evolution of the cluster length and width also exhibit
power-law scaling with universal growth exponents for all considered values of
$F$. We identify a critical cluster length $L_{c}$ that marks a transition from
one-dimensional to self-similar two-dimensional cluster growth. Moreover, we
find that the cluster properties depend markedly on the critical cluster size
$i^{*}$ of the isotropically interacting reference system ($\eta = 1$).
