Causal discovery from observational data, especially for count data, is essential across scientific and industrial contexts, such as biology, economics, and network operation maintenance. For this task, most approaches model count data using Bayesian networks or ordinal relations. However, they overlook the inherent branching structures that are frequently encountered, e.g., a browsing event might trigger an adding cart or purchasing event. This can be modeled by a binomial thinning operator (for branching) and an additive independent Poisson distribution (for noising), known as Poisson Branching Structure Causal Model (PB-SCM). There is a provably sound cumulant-based causal discovery method that allows the identification of the causal structure under a branching structure. However, we show that there still remains a gap in that there exist causal directions that are identifiable while the algorithm fails to identify them. In this work, we address this gap by exploring the identifiability of PB-SCM using the Probability Generating Function (PGF). By developing a compact and exact closed-form solution for the PGF of PB-SCM, we demonstrate that each component in this closed-form solution uniquely encodes a specific local structure, enabling the identification of the local structures by testing their corresponding component appearances in the PGF. Building on this, we propose a practical algorithm for learning causal skeletons and identifying causal directions of PB-SCM using PGF. The effectiveness of our method is demonstrated through experiments on both synthetic and real datasets.