Probabilistic models of allocating shots to boxes according to a certain probability distribution have commonly been used for processes involving agglomeration. Such processes are of interest in many areas of research such as ecology, physiology, chemistry and genetics. Time could be incorporated into the shots-and-boxes model by considering multiple layers of boxes through which the shots move, where the layers represent the passing of time. Such a scheme with multiple layers, each with a certain number of occupied boxes is naturally associated with a random tree. It lends itself to genetic applications where the number of ancestral lineages of a sample changes through the generations. This multiple-layer scheme also allows us to explore the difference in the number of occupied boxes between layers, which gives a measure of how quickly merges are happening. In particular, results for the multiple-layer scheme corresponding to those known for a single-layer scheme, where, under certain conditions, the limiting distribution of the number of occupied boxes is either Poisson or normal, are derived. To provide motivation and demonstrate which methods work well, a detailed study of a small, finite example is provided. A common approach for establishing a limiting distribution for a random variable of interest is to first show that it can be written as a sum of independent Bernoulli random variables as this then allows us to apply standard central limit theorems. Additionally, it allows us to, for example, provide an upper bound on the distance to a Poisson distribution. One way of showing that a random variable can be written as a sum of independent Bernoulli random variables is to show that its probability generating function (p.g.f.) has all real roots. Various methods are presented and considered for proving the p.g.f. of the number of occupied boxes in any given layer of the scheme has all real roots. By considering small finite examples some of these methods could be ruled out for general N. Finally, the scheme for general N boxes and n shots is considered, where again a uniform allocation of shots is used. It is shown that, under certain conditions, the distribution of the number of occupied boxes tends towards either a normal or Poisson limit. Equivalent results are also demonstrated for the distribution of the difference in the number of occupied boxes between consecutive layers.