  Recently, it was shown that fractional quantum Hall states can be defined on
fractal lattices. Proposed exact parent Hamiltonians for these states are
nonlocal and contain three-site terms. In this work, we look for simpler,
approximate parent Hamiltonians for bosonic Laughlin states at half filling,
which contain only onsite potentials and two-site hopping with the interaction
generated implicitly by hardcore constraints (as in the Hofstadter and
Kapit-Mueller models on periodic lattices). We use an ``inverse method'' to
determine such Hamiltonians on finite-generation Sierpi\'{n}ski carpet and
triangle lattices. The ground states of some of the resulting models display
relatively high overlap with the model states if up to third neighbor hopping
terms are considered, and by increasing the maximum hopping distance one can
achieve nearly perfect overlaps. When the number of particles is reduced and
additional potentials are introduced to trap quasiholes, the overlap with a
model quasihole wavefunction is also high in some cases, especially for the
nonlocal Hamiltonians. We also study how the small system size affects the
braiding properties for the model quasihole wavefunctions and perform analogous
computations for Hamiltonian models.
