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Accurately extracting the airway tree is of utmost importance to correctly analyze CT images of the lungs. A survey of published methods reveals the existence of a trade-off between sensitivity -number of airway branches found- and accuracy -how much parenchymal leakage occurs-. In this paper, we present an algorithm for robust airway segmentation that attains both high sensitivity and accuracy. This is accomplished by using an initial permissive voxel acceptance criterion followed by early leakage detection and correction using a novel leakage recovery algorithm. Our algorithm was tested by comparing it to manual segmentation of a large and diverse image data-set.