  Wave dark matter ($\psi \rm{DM}$) composed of extremely light bosons
($m_{\psi} \sim 10^{-22}\,\rm eV$), with quantum pressure suppressing
structures below a kpc-scale de Broglie wavelength, has become a viable dark
matter candidate. Compared to the conventional free-particle $\psi {\rm DM}$
(${\rm FP} \psi {\rm DM}$), the extreme-axion $\psi \rm{DM}$ model (${\rm EA}
\psi {\rm DM}$) proposed by Zhang & Chiueh (2017) features a larger cut-off
wavenumber and a broad spectral bump in the matter transfer function. Here we
conduct cosmological simulations to compare the halo abundances and assembly
histories at $z=4-11$ between three different scenarios: ${\rm FP} \psi {\rm
DM}$, ${\rm EA} \psi {\rm DM}$, and cold dark matter (CDM). We show that ${\rm
EA} \psi {\rm DM}$ produces significantly more abundant low-mass haloes than
${\rm FP} \psi {\rm DM}$ with the same $m_{\psi}$, and therefore could
alleviate the tension in $m_{\psi}$ required by the Ly$\alpha$ forest data and
by the kpc-scale dwarf galaxy cores. We also find that, compared to the CDM
counterparts, massive ${\rm EA} \psi {\rm DM}$ haloes are on average $3-4$
times more massive at $z=10-11$ due to their earlier formation, undergo a
slower mass accretion at $7 \lesssim z \lesssim 11$, and then show a rapidly
rising major merger rate exceeding CDM by $\sim 50\%$ at $4 \lesssim z \lesssim
7$. This fact suggests that ${\rm EA} \psi {\rm DM}$ haloes may exhibit more
prominent starbursts at $z \lesssim 7$.
