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The decision tree is a flexible machine learning model that finds its successin numerous applications. It is usually fitted in a recursively greedy mannerusing CART. In this paper, we investigate the convergence rate of CART under aregression setting. First, we establish an upper bound on the prediction errorof CART under a sufficient impurity decrease (SID) condition\cite{chi2022asymptotic} -- our result improves upon the known result by\cite{chi2022asymptotic} under a similar assumption. Furthermore, we provideexamples that demonstrate the error bound cannot be further improved by morethan a constant or a logarithmic factor. Second, we introduce a set of easilyverifiable sufficient conditions for the SID condition. Specifically, wedemonstrate that the SID condition can be satisfied in the case of an additivemodel, provided that the component functions adhere to a ``locally reversePoincar{\'e} inequality". We discuss several well-known function classes innon-parametric estimation to illustrate the practical utility of this concept.

On the Convergence of CART under Sufficient Impurity Decrease Condition