Tree-based models have been successfully applied to a wide variety of tasks, including time series forecasting. They are increasingly in demand and widely accepted because of their comparatively high level of interpretability. However, many of them suffer from the overfitting problem, which limits their application in real-world decision-making. This problem becomes even more severe in online-forecasting settings where time series observations are incrementally acquired, and the distributions from which they are drawn may keep changing over time. In this context, we propose a novel method for the online selection of tree-based models using the TreeSHAP explainability method in the task of time series forecasting. We start with an arbitrary set of different tree-based models. Then, we outline a performance-based ranking with a coherent design to make TreeSHAP able to specialize the tree-based forecasters across different regions in the input time series. In this framework, adequate model selection is performed online, adaptively following drift detection in the time series. In addition, explainability is supported on three levels, namely online input importance, model selection, and model output explanation. An extensive empirical study on various real-world datasets demonstrates that our method achieves excellent or on-par results in comparison to the state-of-the-art approaches as well as several baselines.