This paper proposes a method to extract vessel trees by continually extending detected branches with locally optimal paths. Our approach uses a cost function from a multiscale vessel enhancement filter. Optimal paths are selected based on rules that take into account the geometric characteristics of the vessel tree. Experiments were performed on 10 low dose chest CT scans for which the pulmonary vessel trees were extracted. The proposed method is shown to extract a better connected vessel tree and extract more of the small peripheral vessels in comparison to applying a threshold on the output of the vessel enhancement filter.