The treatment planning process for patients with head and neck (H&N) cancer is regarded as one of the most complicated due to large target volume, multiple prescription dose levels, and many radiation-sensitive critical structures near the target. Treatment planning for this site requires a high level of human expertise and a tremendous amount of effort to produce personalized high quality plans, taking as long as a week, which deteriorates the chances of tumor control and patient survival. To solve this problem, we propose to investigate a deep learning-based dose prediction model, Hierarchically Densely Connected U-net, based on two highly popular network architectures: U-net and DenseNet. We find that this new architecture is able to accurately and efficiently predict the dose distribution, outperforming the other two models, the Standard U-net and DenseNet, in homogeneity, dose conformity, and dose coverage on the test data. Averaging across all organs at risk, our proposed model is capable of predicting the organ-at-risk max dose within 6.3% and mean dose within 5.1% of the prescription dose on the test data. The other models, the Standard U-net and DenseNet, performed worse, having an averaged organ-at-risk max dose prediction error of 8.2% and 9.3%, respectively, and averaged mean dose prediction error of 6.4% and 6.8%, respectively. In addition, our proposed model used 12 times less trainable parameters than the Standard U-net, and predicted the patient dose 4 times faster than DenseNet.