Edamame is a type of green, vegetable soybean and improving shoot architecture traits for edamame is important for breeding of high-yield varieties by decreasing potential loss due to harvesting. In this study, we use digital imaging technology and computer vision algorithms to characterize major traits of shoot architecture for edamame. Using a population of edamame PIs, we seek to identify underlying genetic control of different shoot architecture traits. We found significant variations in the shoot architecture of the edamame lines including long-skinny and candle stick-like structures. To quantify the similarity and differences of branching patterns between these edamame varieties, we applied a topological measurement called persistent homology. Persistent homology uses algebraic geometry algorithms to measure the structural similarities between complex shapes. We found intriguing relationships between the topological features of branching networks and pod numbers in our plant population, suggesting combination of multiple topological features contribute to the overall pod numbers on a plant. We also identified potential candidate genes including a lateral organ boundary gene family protein and a MADS-box gene that are associated with the pod numbers. This research provides insight into the genetic regulation of shoot architecture traits and can be used to further develop edamame varieties that are better adapted to mechanical harvesting.