Large induced trees in dense random graphs

Erd\H{o}s and Palka initiated the study of the maximal size of induced trees in random graphs in 1983. They proved that for every fixed $0<p<1$ the size of a largest induced tree in $G_{n,p}$ is concentrated around $2{\log }_q(np)$ with high probability, where $q=(1-p{)}^{-1}$. De la Vega showed concentration around the same value for $p=C\mathrm{/}n$ where $C$ is a large constant, and his proof also works for all larger $p$. We show that for any given tree $T$ with bounded maximum degree and of size $(2-o(1)){\log }_q(np)$, $G_{n,p}$ contains an induced copy of $T$ with high probability for $n^{-1\mathrm{/}2}{\ln }^{10\mathrm{/}9}n\le p\le 0.99$. This is asymptotically optimal.