Purpose: To assess whether the three-dimensional (3D) structural configuration of the central retinal vessel trunk and its branches (CRVT&B) could be used as a diagnostic marker for glaucoma. Method: We trained a deep learning network to automatically segment the CRVT&B from the B-scans of the optical coherence tomography (OCT) volume of the optic nerve head (ONH). Subsequently, two different approaches were used for glaucoma diagnosis using the structural configuration of the CRVT&B as extracted from the OCT volumes. In the first approach, we aimed to provide a diagnosis using only 3D CNN and the 3D structure of the CRVT&B. For the second approach, we projected the 3D structure of the CRVT&B orthographically onto three planes to obtain 2D images, and then a 2D CNN was used for diagnosis. The segmentation accuracy was evaluated using the Dice coefficient, whereas the diagnostic accuracy was assessed using the area under the receiver operating characteristic curves (AUC). The diagnostic performance of the CRVT&B was also compared with that of retinal nerve fiber layer (RNFL) thickness. Results: Our segmentation network was able to efficiently segment retinal blood vessels from OCT scans. On a test set, we achieved a Dice coefficient of 0.81\pm0.07. The 3D and 2D diagnostic networks were able to differentiate glaucoma from non-glaucoma subjects with accuracies of 82.7% and 83.3%, respectively. The corresponding AUCs for CRVT&B were 0.89 and 0.90, higher than those obtained with RNFL thickness alone. Conclusions: Our work demonstrated that the diagnostic power of the CRVT&B is superior to that of a gold-standard glaucoma parameter, i.e., RNFL thickness. Our work also suggested that the major retinal blood vessels form a skeleton -- the configuration of which may be representative of major ONH structural changes as typically observed with the development and progression of glaucoma.