Continuous 3+2-axis additive manufacturing (AM) is an emerging AM technology that overcomes the signature problems of the need of support structures and the staircase effect of the traditional 3-axis AM. This paper presents a novel process planning framework for automatically generating a multi-axis support-free printing path for continuous 3+2-axis AM of an arbitrary freeform part. The framework is based on the geometric processing of skeletonization and decomposition and is particularly suitable for a part with distinct multiple trunk-branch structures. The physical printing experimental results carried out by the authors indicate that the proposed framework has performed well and fabricated some challenging models with large overhangs and twisty spatial topologies.