We present new results about fully leafed induced subtrees in Penrose P2 tilings, also known as kites and darts tilings. We first determine the graph structure of these subtrees and show that they are caterpillars up to an appendix of at most six tiles. In other words, if we remove their degree one vertices, then they are path graphs up to an additional connected path of at most two tiles. We then study bi-infinite fully leafed induced caterpillars in P2 tilings and their geometric properties. In particular, we refute the conjecture proposed by C. Porrier, A. Goupil and A. Blondin Massé that there is a unique bi-infinite fully leafed caterpillar in Penrose P2 tilings.