  We analyze the coalescing model where a `primary' colony grows and randomly
emits secondary colonies that spread and eventually coalesce with it. This
model describes population proliferation in theoretical ecology, tumor growth
and is also of great interest for modeling the development of cities. Assuming
the primary colony to be always spherical of radius $r(t)$ and the emission
rate proportional to $r(t)^\theta$ where $\theta>0$, we derive the mean-field
equations governing the dynamics of the primary colony, calculate the scaling
exponents versus $\theta$ and compare our results with numerical simulations.
We then critically test the validity of the circular approximation and show
that it is sound for a constant emission rate ($\theta=0$). However, when the
emission rate is proportional to the perimeter, the circular approximation
breaks down and the roughness of the primary colony can not be discarded, thus
modifying the scaling exponents.
