We propose a deep reinforcement learning-based solution for the 3D reconstruction of objects of complex topologies from a single RGB image. We use a template-based approach. However, unlike previous template-based methods, which are limited to the reconstruction of 3D objects of fixed topology, our approach learns simultaneously the geometry and topology of the target 3D shape in the input image. To this end, we propose a neural network that learns to deform a template to fit the geometry of the target object. Our key contribution is a novel reinforcement learning framework that enables the network to also learn how to adjust, using pruning operations, the topology of the template to best fit the topology of the target object. We train the network in a supervised manner using a loss function that enforces smoothness and penalizes long edges in order to ensure high visual plausibility of the reconstructed 3D meshes. We evaluate the proposed approach on standard benchmarks such as ShapeNet, and in-the-wild using unseen real-world images. We show that the proposed approach outperforms the state-of-the-art in terms of the visual quality of the reconstructed 3D meshes, and also generalizes well to out-of-category images.