A cactus tree is a simple data structure that represents all minimum cuts of a weighted graph in linear space. We describe the first algorithm that can build a cactus tree from the asymptotically fastest deterministic algorithm that finds all minimum cuts in a weighted graph -- the Hao-Orlin minimum cut algorithm. This improves the time to construct the cactus in graphs with n vertices and m edges from O(n³) to O(nm log n²/m).