  We prove that a.a.s. the maximum size of an induced subtree of the binomial
random graph $G(n,p)$ is concentrated in 2 consecutive points. We also prove
that, given a non-negative integer-valued function $t(k)<\varepsilon k^2$,
under a certain smoothness condition on this function, a.a.s. the maximum size
$k$ of an induced subgraph with exactly $t(k)$ edges of $G(n,p)$ is
concentrated in 2 consecutive points as well.
