In a recent article by C. Porrier, A. Blondin Massé and A. Goupil, a first bi-infinite fully leafed induced subcaterpillar of Penrose P2 tilings is presented. In this paper, we formally construct this caterpillar for the first time. We then prove that every fully leafed induced subtree in Penrose P2 tilings is a caterpillar with at most one appendix of at most two internal tiles, and we characterize fully leafed induced subtrees that have the property of saturation. We also refute the conjecture that there is a unique bi-infinite fully leafed induced subcaterpillar by constructing a new one. Finally, we present progress on the construction of all bi-infinite fully leafed induced subcaterpillars in Penrose P2 tilings.