  We prove that every triangle-free planar graph of order $n$ and size $m$ has
an induced linear forest with at least $\frac{9n - 2m}{11}$ vertices, and thus
at least $\frac{5n + 8}{11}$ vertices. Furthermore, we show that there are
triangle-free planar graphs on $n$ vertices whose largest induced linear forest
has order $\lceil \frac{n}{2} \rceil + 1$.
