Maximum induced tree
This is $\mathrm{NP}$-complete. Reduce from Independent Set instance $(G, k)$. Add a new universal vertex $v$ to $G$. Connect $v$ with a very long path (only to $v$ not to any original vertex of $G$). And set $k' = k + 1 + \mathrm{length\:of\:the\:long\:path}$, where length of the long path is the number of vertices in it.

Clearly, if the induced tree does not touch $v$ then it either only stucks in the long path or being confined to $V$. So, in order to have $k'$ vertices, it must include $v$ and the long path. Since $v$ is connected to all $V$, the set of vertices in $V$ that are included the induced tree must form an independent set.