Robust and accurate segmentation of the human airway tree from multi-detector computed-tomography (MDCT) chest scans is vital for many pulmonary-imaging applications. As modern MDCT scanners can detect hundreds of airway tree branches, manual segmentation and semi-automatic segmentation requiring significant user intervention are impractical for producing a full global segmentation. Fully-automated methods, however, may fail to extract small peripheral airways. We propose an automatic algorithm that searches the entire lung volume for airway branches and poses segmentation as a global graph-theoretic optimization problem. The algorithm has shown strong performance on 23 human MDCT chest scans acquired by a variety of scanners and reconstruction kernels. Visual comparisons with adaptive region-growing results and quantitative comparisons with manually-defined trees indicate a high sensitivity to peripheral airways and a low false-positive rate. In addition, we propose a suite of interactive segmentation tools for cleaning and extending critical areas of the automatically segmented result. These interactive tools have potential application for image-based guidance of bronchoscopy to the periphery, where small, terminal branches can be important visual landmarks. Together, the automatic segmentation algorithm and interactive tool suite comprise a robust system for human airway-tree segmentation.