  Hydrogen-rich supernovae, known as Type II (SNe II), are the most common
class of explosions observed following the collapse of the core of massive
stars. We use analytical estimates and population synthesis simulations to
assess the fraction of SNe II progenitors that are expected to have exchanged
mass with a companion prior to explosion. We estimate that 1/3 to 1/2 of SN II
progenitors have a history of mass exchange with a binary companion before
exploding. The dominant binary channels leading to SN II progenitors involve
the merger of binary stars. Mergers are expected to produce a diversity of SN
II progenitor characteristics, depending on the evolutionary timing and
properties of the merger. Alternatively, SN II progenitors from interacting
binaries may have accreted mass from their companion, and subsequently been
ejected from the binary system after their companion exploded. We show that the
overall fraction of SN II progenitors that are predicted to have experienced
binary interaction is robust against the main physical uncertainties in our
models. However, the relative importance of different binary evolutionary
channels is affected by changing physical assumptions. We further discuss ways
in which binarity might contribute to the observed diversity of SNe II by
considering potential observational signatures arising from each binary
channel. For supernovae which have a substantial H-rich envelope at explosion
(i.e., excluding Type IIb SNe), a surviving non-compact companion would
typically indicate that the supernova progenitor star was in a wide,
non-interacting binary. We argue that a significant fraction of even Type II-P
SNe are expected to have gained mass from a companion prior to explosion.
