Accurate segmentation of the pulmonary airway tree is crucial for diagnosing lung diseases. To tackle the issues of low segmentation accuracy and frequent leaks in existing methods, this paper proposes a precise segmentation method using quasi-spherical region-constrained wavefront propagation with tracheal wall gap sealing. Based on the characteristic that the surface formed by seed points approximates the airway cross-section, the width of the unsegmented airway is calculated, determining the initial quasi-spherical constraint region. Using the wavefront propagation method, seed points are continuously propagated and segmented along the tracheal wall within the quasi-spherical constraint region, thus overcoming the need to determine complex segmentation directions. To seal tracheal wall gaps, a morphological closing operation is utilized to extract the characteristics of small holes and locate low-brightness tracheal wall gaps. By filling the CT values at these gaps, the method seals the tracheal wall gaps. Extensive experiments on the EXACT09 dataset demonstrate that our algorithm ranks third in segmentation completeness. Moreover, its performance in preventing airway leaks is significantly better than the top-two algorithms, effectively preventing large-scale leak-induced spread.