Automatic lung organ segmentation on CT images is crucial for lung disease diagnosis. However, the unlimited voxel values and class imbalance of lung organs can lead to false-negative/positive and leakage issues in advanced methods. Additionally, some slender lung organs are easily lost during the recycled down/up-sample procedure, e.g., bronchioles & arterioles, causing severe discontinuity issue. Inspired by these, this paper introduces an effective lung organ segmentation method called Fuzzy Attention-based Border Rendering (FABR) network. Since fuzzy logic can handle the uncertainty in feature extraction, hence the fusion of deep networks and fuzzy sets should be a viable solution for better performance. Meanwhile, unlike prior top-tier methods that operate on all regular dense points, our FABR depicts lung organ regions as cube-trees, focusing only on recycle-sampled border vulnerable points, rendering the severely discontinuous, false-negative/positive organ regions with a novel Global-Local Cube-tree Fusion (GLCF) module. All experimental results, on four challenging datasets of airway & artery, demonstrate that our method can achieve the favorable performance significantly.