Formant tracking is investigated in this study by using trackers based on dynamic programming (DP) and deep neural nets (DNNs). Using the DP approach, six formant estimation methods were first compared. The six methods include linear prediction (LP) algorithms, weighted LP algorithms and the recently developed quasi-closed phase forward-backward (QCP-FB) method. QCP-FB gave the best performance in the comparison. Therefore, a novel formant tracking approach, which combines benefits of deep learning and signal processing based on QCP-FB, was proposed. In this approach, the formants predicted by a DNN-based tracker from a speech frame are refined using the peaks of the all-pole spectrum computed by QCP-FB from the same frame. Results show that the proposed DNN-based tracker performed better both in detection rate and estimation error for the lowest three formants compared to reference formant trackers. Compared to the popular Wavesurfer, for example, the proposed tracker gave a reduction of 29%, 48% and 35% in the estimation error for the lowest three formants, respectively.