We propose a method for transfer learning in nonparametric regression using a random forest (RF) with distance covariance-based feature weights, assuming the unknown source and target regression functions are sparsely different. Our method obtains residuals from a source domain-trained Centered RF (CRF) in the target domain, then fits another CRF to these residuals with feature splitting probabilities proportional to feature-residual sample distance covariance. We derive an upper bound on the mean square error rate of the procedure as a function of sample sizes and difference dimension, theoretically demonstrating transfer learning benefits in random forests. A major difficulty for transfer learning in random forests is the lack of explicit regularization in the method. Our results explain why shallower trees with preferential selection of features lead to both lower bias and lower variance for fitting a low-dimensional function. We show that in the residual random forest, this implicit regularization is enabled by sample distance covariance. In simulations, we show that the results obtained for the CRFs also hold numerically for the standard RF (SRF) method with data-driven feature split selection. Beyond transfer learning, our results also show the benefit of distance-covariance-based weights on the performance of RF when some features dominate. Our method shows significant gains in predicting the mortality of ICU patients in smaller-bed target hospitals using a large multi-hospital dataset of electronic health records for 200,000 ICU patients.