In this paper, we propose a supervised domain adaptation (DA) framework for adapting decision forests in the presence of distribution shift between training (source) and testing (target) domains, given few labeled examples. We introduce a novel method for DA through an error-correcting hierarchical transfer relaxation scheme with domain alignment, feature normalization, and leaf posterior reweighting to correct for the distribution shift between the domains. For the first time we apply DA to the challenging problem of extending in vitro trained forests (source domain) for in vivo applications (target domain). The proof-of-concept is provided for in vivo characterization of atherosclerotic tissues using intravascular ultrasound signals, where presence of flowing blood is a source of distribution shift between the two domains. This potentially leads to misclassification upon direct deployment of in vitro trained classifier, thus motivating the need for DA as obtaining reliable in vivo training labels is often challenging if not infeasible. Exhaustive validations and parameter sensitivity analysis substantiate the reliability of the proposed DA framework and demonstrates improved tissue characterization performance for scenarios where adaptation is conducted in presence of only a few examples. The proposed method can thus be leveraged to reduce annotation costs and improve computational efficiency over conventional retraining approaches.