This letter introduces a generalization of Isolation Forest (IF) based on the existing Extended IF (EIF). EIF has shown some interest compared to IF being for instance more robust to some artefacts. However, some information can be lost when computing the EIF trees since the sampled threshold might lead to empty branches. This letter introduces a generalized isolation forest algorithm called Generalized IF (GIF) to overcome these issues. GIF is faster than EIF with a similar performance, as shown in several simulation results associated with reference databases used for anomaly detection.