Earthquake occurrences are often described using a class of branching models called Epidemic-Type Aftershock Sequence (ETAS) models. The name derives from the fact that the model allows earthquakes to cause aftershocks, and then those aftershocks may induce subsequent aftershocks, and so on. Despite their value in seismology, such models have not previously been used in studying the incidence of invasive plant and animal species. Here, we apply a modified version of the space-time ETAS model to study the spread of an invasive species red banana trees (Musa velutina) in a Costa Rican rainforest. One challenge in this ecological application is that fitting the model requires the originations of the plants, which are not observed but may be estimated using filed data on the heights of the plants on a given date and their empirical growth rates. The formulation of the triggering density function, which describes the way events cause future occurrences of events is based on plots of inter-event times and distances for the red banana plants. We characterize the estimated spatial-temporal rate of spread of red banana plants using a space-time ETAS model. We then assess the triggering density more carefully using a non-parametric stochastic declustering method based on Marsan and Lengliné (2008). When the algorithm is applied to the red banana data, the results indicate similar temporal and spatial structure, compared to previous estimates, as well as triggering of offspring running primarily to the northwest and the southeast from each parent. Non-parametric results are also used to obtain estimates of the most likely targets where immigration of red banana plants may be occurring.