Birth-death processes form a natural class where ideas and results on large deviations can be tested. In this paper, we derive a large deviation principle under the assumption that the rate of a jump down (death) is growing asymptotically linearly with the population size, while the rate of a jump up (birth) is growing sub-linearly. We establish a large deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large deviation probabilities. The results show interesting features of dependence of the large deviation functional upon the parameters of the process and the forms of scaling and normalization.