We present a method of selecting quasars up to redshift approximate to 6 with random forests, a supervised machine-learning method, applied to Pan-STARRS1 and WISE data. We find that, thanks to the increasing set of known quasars, we can assemble a training set that enables supervised machine-learning algorithms to become a competitive alternative to other methods up to this redshift. We present a candidate set for the redshift range 4.8-6.3, which includes the region around z = 5.5 where selecting quasars is difficult due to their photometric similarity to red and brown dwarfs. We demonstrate that, under our survey restrictions, we can reach a high completeness (66% +/- 7% below redshift 5.6/83(-9)(+6)% above redshift 5.6) while maintaining a high selection efficiency (78(-8)(+10)%/94(-8)(+5)%). Our selection efficiency is estimated via a novel method based on the different distributions of quasars and contaminants on the sky. The final catalog of 515 candidates includes 225 known quasars. We predict the candidate catalog to contain additional 148(-33)(+41) new quasars below redshift 5.6 and 45(-8)(+5) above, and we make the catalog publicly available. Spectroscopic follow-up observations of 37 candidates led us to discover 20 new high redshift quasars (18 at 4.6 <= z <= 5.5, 2 z similar to 5.7). These observations are consistent with our predictions on efficiency. We argue that random forests can lead to higher completeness because our candidate set contains a number of objects that would be rejected by common color cuts, including one of the newly discovered redshift 5.7 quasars.