  A common subgraph of two graphs $G_1$ and $G_2$ is a graph that is isomorphic
to subgraphs of $G_1$ and $G_2$. In the largest common subgraph problem the
task is to determine a common subgraph for two given graphs $G_1$ and $G_2$
that is of maximum possible size ${\rm lcs}(G_1,G_2)$. This natural problem
generalizes the well-studied graph isomorphism problem, has many applications,
and remains NP-hard even restricted to unions of paths. We present a simple
$4$-approximation algorithm for forests, and, for every fixed $\epsilon\in
(0,1)$, we show that, for two given forests $F_1$ and $F_2$ of order at most
$n$, one can determine in polynomial time a common subgraph $F$ of $F_1$ and
$F_2$ with at least ${\rm lcs}(F_1,F_2)-\epsilon n$ edges. Restricted to
instances with ${\rm lcs}(F_1,F_2)\geq cn$ for some fixed positive $c$, this
yields a polynomial time approximation scheme. Our approach relies on the
approximation of the given forests by structurally simpler forests that are
composed of copies of only $O(\log (n))$ different starlike rooted trees and
iterative quantizations of the options for the solutions.
