  Numerous observations have shown that almost all galaxies in our Universe
host supermassive black holes (SMBHs), but there is still much debate about
their formation and evolutionary processes. Recently, gravitational waves (GWs)
have been expected to be a new and important informative observation, in
particular, in the low-frequency region by making use of the Laser
Interferometer Space Antenna (LISA) and Pulsar Timing Arrays (PTAs). As an
evolutionary process of the SMBHs, we revisit a dark matter (DM) halo-SMBH
coevolution model based on the halo merger tree employing an ansatz for the
mass relation between the DM halos and the SMBHs at $z=6$. In this model, the
mass of SMBHs grows through their mergers associated with the halo mergers, and
hence the evolutionary information must be stored in the GWs emitted at the
mergers. We investigate the stochastic gravitational background from the
coalescing SMBH binaries, which the PTAs can detect, and also the GW bursts
emitted at the mergers, which can be detected by the mHz band observations such
as LISA. We also discuss the possibility of probing the mass relation between
the DM halos and the SMBHs at high redshift by future GW observations.
