  We consider non-compact branes in topological string theories on a class of
Calabi-Yau spaces including the resolved conifold and its mirror. We compute
the amplitudes of the insertion of non-compact Lagrangian branes in the A-model
on the resolved conifold in the context of the topological vertex as well as
the melting crystal picture. They all agree with each other and also agree with
the results from Chern-Simons theory, supporting the large N duality. We find
that they obey the Schr\"odinger equation confirming the wavefunction behavior
of the amplitudes. We also compute the amplitudes of the non-compact B-branes
in the DV matrix model which arises as a B-model open string field theory on
the mirror manifold of the deformed conifold. We take the large N duality to
consider the B-model on the mirror of the resolved conifold and confirm the
wave function behavior of this amplitude. We find appropriate descriptions of
non-compact branes in each model, which give complete agreements among those
amplitudes and clarify the salient features including the role of symmetries
toward these agreements.
