  Statistical analysis is increasingly confronted with complex data from metric
spaces. Petersen and M\"uller (2019) established a general paradigm of
Fr\'echet regression with complex metric space valued responses and Euclidean
predictors. However, the local approach therein involves nonparametric kernel
smoothing and suffers from the curse of dimensionality. To address this issue,
we in this paper propose a novel random forest weighted local Fr\'echet
regression paradigm. The main mechanism of our approach relies on a locally
adaptive kernel generated by random forests. Our first method uses these
weights as the local average to solve the conditional Fr\'echet mean, while the
second method performs local linear Fr\'echet regression, both significantly
improving existing Fr\'echet regression methods. Based on the theory of
infinite order U-processes and infinite order $M_{m_n}$-estimator, we establish
the consistency, rate of convergence, and asymptotic normality for our local
constant estimator, which covers the current large sample theory of random
forests with Euclidean responses as a special case. Numerical studies show the
superiority of our methods with several commonly encountered types of responses
such as distribution functions, symmetric positive-definite matrices, and
sphere data. The practical merits of our proposals are also demonstrated
through the application to New York taxi data and human mortality data.
