The largest common subtree of two random trees

We study the size and structure of the largest common subtree (LCS) between two independent Bienaym\'e trees conditioned to have size $n$. When the trees are critical with finite $2$nd and $(2+\kappa)$th moment respectively for some $\kappa>0$, we prove that the LCS has size of order $\sqrt{n}$, and is approximated by the length of three paths meeting at a central node. Moreover, we show that the largest common subtree between two critical independent Bienaym\'e trees with size $n$ and finite second moments may be much larger than $\sqrt{n}$, implying that our result is tight. We also pose a number of open questions and suggestions for future research.