  The decycling number of a graph $G$ is the minimum number of vertices whose
removal from $G$ results in an acyclic subgraph. It is known that determining
the decycling number of a graph $G$ is equivalent to finding the maximum
induced forests of $G$. The line graphs of trees are the claw-free block
graphs. These graphs have been used by Erd\H{o}s, Saks and S\'{o}s to construct
graphs with a given number of edges and vertices whose maximum induced tree is
very small. In this paper, we give bounds on the decycling number of line
graphs of trees and construct extremal trees to show that these bounds are the
best possible. We also give bounds on the decycling number of line graph of
$k$-ary trees and determine the exact the decycling number of line graphs of
perfect $k$-ary trees.
