As a flexible nonparametric learning tool, random forest has been widely applied to various real applications with appealing empirical performance, even in the presence of high-dimensional feature space. Unveiling the underlying mechanisms has led to some important recent theoretical results on consistency under the classical setting of fixed dimensionality or for some modified version of the random forest algorithm. Yet the consistency rates of the original version of the random forest algorithm in a general high-dimensional nonparametric regression setting remain largely unexplored. In this paper, we fill such a gap and build a high-dimensional consistency theory for random forest. Our new theoretical results show that random forest can indeed adapt to high dimensions and also provide some insights into the role of sparsity from the perspective of feature relevance.