  The stationary asymptotic properties of the diffusion limit of a multi-type
branching process with neutral mutations are studied. For the critical and
subcritical processes the interesting limits are those of quasi-stationary
distributions conditioned on non-extinction. Pedagogical derivations are given
for known results that the limiting distributions for supercritical and
critical processes are found to collapse onto rays aligned with stationary
eigenvectors of the mutation rate matrix, in agreement with discrete multi-type
branching processes. For the sub-critical process the previously unsolved
quasi-stationary distribution is obtained to first order in the overall
mutation rate, which is assumed to be small. The sampling distribution over
allele types for a sample of given finite size is found to agree to first order
in mutation rates with the analogous sampling distribution for a Wright-Fisher
diffusion with constant population size.
