  Halo merger trees describe the hierarchical mass assembly of dark matter
haloes, and are the backbone for modeling galaxy formation and evolution.
Merger trees constructed using Monte Carlo algorithms based on the extended
Press-Schechter (EPS) formalism are complementary to those extracted from
N-body simulations, and have the advantage that they are not trammeled by
limited numerical resolution and uncertainties in identifying (sub)haloes and
linking them between snapshots. This paper compares multiple EPS-based merger
tree algorithms to simulation results using four diagnostics: progenitor mass
function (PMF), mass assembly history (MAH), merger rate per descendant halo,
and the unevolved subhalo mass function (USMF). In general, algorithms based on
spherical collapse yield major-merger rates that are too high by a factor of
two, resulting in MAHs that are systematically offset. Assuming ellipsoidal
collapse solves most of these issues, but the particular algorithm investigated
here that incorporates ellipsoidal collapse dramatically overpredicts the
minor-merger rate for massive haloes. The only algorithm in our comparison that
yields MAHs, merger rates, and USMFs in good agreement with simulations, is
that by Parkinson et al. (2008). However, this is not a true EPS-based
algorithm as it draws its progenitor masses from a PMF calibrated against
simulations, rather than `predicted' by EPS. Finally we emphasize that the
benchmarks used to test the EPS algorithms are obtained from simulations and
are hampered by significant uncertainties themselves. In particular, MAHs and
halo merger rates obtained from simulations by different authors reveal
discrepancies that easily exceed 50 percent, even when based on the same
simulation. Given this status quo, merger trees constructed using the Parkinson
et al. algorithm are as accurate as those extracted from N-body simulations.

Cosmology and Nongalactic Astrophysics