## 1 Introduction

## 2 Method

^{1}(which can also be important for convergence in galaxy formation models; Benson et al. 2012).

^{2}to predict properties of galaxies which form in augmented and unaugmented merger trees in order to explore the how well convergence in galaxy properties is achieved using our augmenting procedure.

### 2.1 Application to Millennium simulation trees

^{3}(Lemson and The Virgo Consortium 2006). We choose \(M_{\mathrm{cut}}=7.08\times10^{10}{\mathrm{M}}_{\odot}\), corresponding to 60 particles. We find that using a lower \(M_{\mathrm{cut}}\) results in incorrect progenitor mass functions in the augmented trees, suggesting that these masses of lower mass halos are insufficiently reliable for our purposes. We choose an initial value for \(\epsilon= \epsilon_{0} \equiv0.15\) by default - we explore in Section 3.1 how sensitive the results are to the choice of this parameter. If a matched tree is not found after \(N_{\mathrm{t}}=50\) trials, we increase \(\epsilon\rightarrow\epsilon( 1 + \epsilon_{0} )\) and continue until a match is found.

^{4}The speed of our algorithm for a given level of convergence will be determined by the interplay of \(\epsilon_{0}\) and \(N_{\mathrm{t}}\). We explore how often ϵ must be increased to find a match in Section 3.1. We augment these trees to a variety of mass resolutions.

## 3 Results

### 3.1 Numerical convergence

^{5}is 19.6, 33.8, 52.3 for \(N_{\mathrm{t}}=50, 100, 200\) respectively. Since each trial is independent, if the probability for any given tree to match successfully is p, then the probability distribution for a match after n trials is simply \(f(n)=p(1-p)^{n-1}\). The mean number of trials, \(\langle f(n) \rangle\) is then approximately consistent with the above results for \(N_{\mathrm{t}}=50, 100, \mbox{and }200\) if \(p=0.0167\). For \(\epsilon_{0}=0.0375\) we find \(p\approx0.0067\). Based on these results, it is clear that \(N_{\mathrm{t}}\) could be increased without significant loss of speed, and with some improvement in accuracy. Specifically, for \(\epsilon_{0}=0.15\) and the specific tree resolutions considered in these tests, \(N_{\mathrm{t}}=50\) will result in around 55% of two-or-more progenitor branches being matched at the original tolerance, \(\epsilon_{0}\), while to have 90% of such branches matched at the original tolerance would required \(N_{\mathrm{t}}\approx 140\). As we will show below, \(N_{\mathrm{t}}=50\) is sufficient to achieve good convergence in all tests that we consider.