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In my understanding, a model tree recursively partitions a dataset, and then uses a linear regression model at each leaf node. On the other hand, a regression spline adds various piecewise polynomials together to reach a final model.

Assuming that the regression spline is adding together linear models, what is the difference between the two types of models?

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In the beginning tree models used regression with a constant in each region (mean, median) and this is the general usage even today. Various extensions were developed, including linear models, oblique trees and so on. For linear models, note also that in general the model were fit only locally to the region at hand, with no constraints regarding models from other regions. Regression splines on the other hand have the requirement that they are continuous. So for a regression spline (which includes MARS) you have additional restrictions to make the regression fit continuous. A second difference is that usually the decision trees uses numerical features only once. Of course this can be modified easily.

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