  Dominant decay of a SM-like Higgs boson into particles beyond those contained
in the minimal supersymmetric standard model has been identified as a natural
scenario to avoid fine tuning in electroweak symmetry breaking while satisfying
all LEP limits. In the simplest such extension, the next-to-minimal
supersymmetric model, the lightest CP-even Higgs boson can decay into two
pseudoscalars. In the scenario with least fine tuning the lightest CP-even
Higgs boson has mass of order 100 GeV. In order to escape LEP limits it must
decay to a pair of the lightest CP-odd Higgs bosons with Br(h -> aa)>.7 and
m_a<2m_b (so that a -> \tau^+ \tau^- or light quarks and gluons). The mass of
the lightest CP-odd Higgs boson is controlled by the soft-trilinear couplings,
A_\lambda(m_Z) and A_\kappa(m_Z). We identify the region of parameter space
where this situation occurs and discuss how natural this scenario is. It turns
out that in order to achieve m_a < 2 m_b with A_\lambda(m_Z), A_\kappa(m_Z) of
order the typical radiative corrections, the required tuning of trilinear
couplings need not be larger than 5-10 %. Further, the necessity for this
tuning can be eliminated in specific SUSY breaking scenarios. Quite
interestingly, Br(h -> aa) is typically above 70 % in this region of parameter
space and thus an appropriately large value requires no additional tuning.
