In this paper we show the occurrence of cubic-root asymptotics in misspecified conditional quantile models where the approximating functions are restricted to be binary decision trees. Inference procedure for the optimal split point in the decision tree is conducted by inverting a t-test or a deviation measure test, both involving Chernoff type limiting distributions. In order to avoid estimating the nuisance parameters in the complicated limiting distribution, subsampling is proved to deliver the correct confidence interval/set.