  We study a graph-theoretic problem in the Penrose P2-graphs which are the
dual graphs of Penrose tilings by kites and darts. Using substitutions, local
isomorphism and other properties of Penrose tilings, we construct a family of
arbitrarily large induced subtrees of Penrose graphs with the largest possible
number of leaves for a given number $n$ of vertices. These subtrees are called
fully leafed induced subtrees. We denote their number of leaves $L_{P2}(n)$ for
any non-negative integer $n$, and the sequence
$\left(L_{P2}(n)\right)_{n\in\mathbb{N}}$ is called the leaf function of
Penrose P2-graphs. We present exact and recursive formulae for $L_{P2}(n)$, as
well as an infinite sequence of fully leafed induced subtrees, which are
caterpillar graphs. In particular, our proof relies on the construction of a
finite graded poset of 3-internal-regular subtrees.
