  We study the fractal and multifractal properties (i.e. the generalized
dimensions of the harmonic measure) of a 2-parameter family of growth patterns
that result from a growth model that interpolates between Diffusion Limited
Aggregation (DLA) and Laplacian Growth Patterns in 2-dimensions. The two
parameters are \beta which determines the size of particles accreted to the
interface, and C which measures the degree of coverage of the interface by each
layer accreted to the growth pattern at every growth step. DLA and Laplacian
Growth are obtained at \beta=0, C=0 and \beta=2, C=1, respectively. The main
purpose of this paper is to show that there exists a line in the \beta-C phase
diagram that separates fractal (D<2) from non-fractal (D=2) growth patterns.
Moreover, Laplacian Growth is argued to lie in the non-fractal part of the
phase diagram. Some of our arguments are not rigorous, but together with the
numerics they indicate this result rather strongly. We first consider the
family of models obtained for \beta=0, C>0, and derive for them a scaling
relation D=2 * D_3. We then propose that this family has growth patterns for
which D=2 for some C>C_{cr}, where C_{cr} may be zero. Next we consider the
whole \beta-C phase diagram and define a line that separates 2-dimensional
growth patterns from fractal patterns with D<2. We explain that Laplacian
Growth lies in the region belonging to 2-dimensional growth patterns,
motivating the main conjecture of this paper, i.e. that Laplacian Growth
patterns are 2-dimensional. The meaning of this result is that the branches of
Laplacian Growth patterns have finite (and growing) area on scales much larger
than any ultra-violet cut-off length.
