  Using stochastic conformal mappings we study the effects of anisotropic
perturbations on diffusion limited aggregation (DLA) in two dimensions. The
harmonic measure of the growth probability for DLA can be conformally mapped
onto a constant measure on a unit circle. Here we map $m$ preferred directions
for growth of angular width $\sigma$ to a distribution on the unit circle which
is a periodic function with $m$ peaks in $[-\pi, \pi)$ such that the width
$\sigma$ of each peak scales as $\sigma \sim 1/\sqrt{k}$, where $k$ defines the
``strength'' of anisotropy along any of the $m$ chosen directions. The two
parameters $(m,k)$ map out a parameter space of perturbations that allows a
continuous transition from DLA (for $m=0$ or $k=0$) to $m$ needle-like fingers
as $k \to \infty$. We show that at fixed $m$ the effective fractal dimension of
the clusters $D(m,k)$ obtained from mass-radius scaling decreases with
increasing $k$ from $D_{DLA} \simeq 1.71$ to a value bounded from below by
$D_{min} = 3/2$. Scaling arguments suggest a specific form for the dependence
of the fractal dimension $D(m,k)$ on $k$ for large $k$, form which compares
favorably with numerical results.
