  We present a simple and efficient empirical algorithm for constructing
dark-matter halo merger trees that reproduce the distribution of trees in the
Millennium cosmological $N$-body simulation. The generated trees are
significantly better than EPS trees. The algorithm is Markovian, and it
therefore fails to reproduce the non-Markov features of trees across short time
steps, except for an accurate fit to the evolution of the average main
progenitor. However, it properly recovers the full main progenitor distribution
and the joint distributions of all the progenitors over long-enough time steps,
$\Delta \omega \simeq \Delta z>0.5$, where $\omega \simeq 1.69/D(t)$ is the
self-similar time variable and $D(t)$ refers to the linear growth of density
fluctuations. We find that the main progenitor distribution is log-normal in
the variable $\sigma^2(M)$, the variance of linear density fluctuations in a
sphere encompassing mass $M$. The secondary progenitors are successfully drawn
one by one from the remaining mass using a similar distribution function. These
empirical findings may be clues to the underlying physics of merger-tree
statistics. As a byproduct, we provide useful, accurate analytic time-invariant
approximations for the main progenitor accretion history and for halo merger
rates.
