  This paper revisits an adaptation of the random forest algorithm for
Fr\'echet regression, addressing the challenge of regression in the context of
random objects in metric spaces. Recognizing the limitations of previous
approaches, we introduce a new splitting rule that circumvents the
computationally expensive operation of Fr\'echet means by substituting with a
medoid-based approach. We validate this approach by demonstrating its
asymptotic equivalence to Fr\'echet mean-based procedures and establish the
consistency of the associated regression estimator. The paper provides a sound
theoretical framework and a more efficient computational approach to Fr\'echet
regression, broadening its application to non-standard data types and complex
use cases.
