  High-redshift luminous quasars powered by accreting supermassive black holes
(SMBHs) with mass $\gtrsim 10^9 M_\odot$ constrain their formation pathways. We
investigate the formation of heavy seeds of SMBHs through gas collapse in the
quasar host progenitors, using merger trees to trace the halo growth in
highly-biased, overdense regions of the universe. The progenitor halos are
likely irradiated by intense H$_2$-photodissociating radiation from nearby
star-forming galaxies and heat the interior gas by successive mergers. The
kinetic energy of the gas originating from mergers as well as baryonic
streaming motion prevents gas collapse and delays prior star formation. With a
streaming velocity higher than the root-mean-square value, gas clouds in nearly
all $10^4$ realizations of merger trees enter the atomic-cooling stage and
begin to collapse isothermally with $T \simeq 8000 K$ via Ly$\alpha$ cooling.
The fraction of trees which host isothermal gas collapse is $14\%$ and
increases with streaming velocity, while the rest form H$_2$-cooled cores after
short isothermal phases. If the collapsing gas is enriched to $Z_{crit}\sim
2\times 10^{-3} Z_\odot$, requiring efficient metal mixing, this fraction could
be reduced by additional cooling via metal fine-structure lines. In the massive
collapsing gas, the accretion rate onto a newly-born protostar ranges between
$3 \times 10^{-3}-5 M_\odot yr^{-1}$, among which a large fraction exceeds the
critical rate suppressing stellar radiative feedback. As a result, we expect a
distribution of stellar mass (presumably BH mass) ranging from several hundred
to above $10^5 M_\odot$, potentially forming massive BH binary mergers and
yielding gravitational wave events.
