Non-volatile processors are proposed for ambient energy harvesting systems to enable accumulative computing across power failures. They employ nonvolatile memory for processor status backup before power outage and resume the system after power recovers. A straightforward backup policy is to back up all volatile data in processors, but it induces high backup cost. In this paper, we focus on branch predictor, an important component in processor, and propose efficient backup schemes to reduce backup cost while maintaining its prediction ability. We first analyze the modules in both traditional and artificial intelligence (AI) assisted designs of branch predictor, and accordingly propose three backup mechanisms pertaining to saturation-driven, locality-driven and maturity-driven backup. On the basis of these mechanisms, adaptive backup branch predictors are designed. Evaluation shows that, with traditional Tournament architecture, the proposed design achieves 15.9% and 54.1% energy reduction when compared with no-backup and all-backup strategy. For AI assisted branch predictor, the proposed design achieves 27.5% and 82.2% energy saving.