The forest of merger history trees associated with the formation of dark matter haloes

We describe a simple, efficient algorithm that allows one to construct Monte Carlo realizations of merger histories of dark matter haloes. The algorithm is motivated by the excursion set model for the conditional and unconditional halo mass functions. The forest of trees constructed using this algorithm depends on the underlying power spectrum. For Poisson or white‐noise initial power spectra, the forest has exactly the same properties as the ensemble of trees described by Sheth. In this case, many ensemble‐averaged higher order statistics of the tree distribution can be computed analytically. For Gaussian initial conditions with more general power spectra, mean properties of our forests closely resemble the mean properties expected from the excursion set approach. For these more general initial conditions, our algorithm shows how to write down simple, analytic approximations to some higher order statistical quantities associated with the forest. These higher order statistics generated using our algorithm, and the associated analytic approximations, are in good agreement with what is measured in numerical simulations of hierarchical gravitational clustering.