Asymptotic order of the largest common subtree for uniform attachment trees

Determine the order of magnitude of the largest common subtree size X_n for two independent uniform attachment trees, including whether X_n=n^{1-o(1)} with high probability and, if so, the finer asymptotics in the exponent.

Background

The paper studies X_n, the size of the largest common subtree of two independent unlabeled uniform attachment trees with n vertices each. It proves a high-probability lower bound of n{0.83}, upper bounds showing that X_n is neither asymptotically equal to n nor uniformly bounded away from zero as a fraction of n, and conjectures near-linear growth in the sense X_n=n{1-o(1)} with high probability.

The precise asymptotic scale remains unresolved. The authors also identify stronger quantitative lower bounds and sharper asymptotics as natural refinements of the main conjecture.

References

We leave as an intriguing open question to understand the magnitude of the size of the largest common subtree.

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Abstract; Section 1, Subsection 1.3, item “Asymptotics of X_n”

In a different direction, is it true that $X_{n} = o(n)$ with high probability?

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Section 1, Subsection 1.3, item “Asymptotics of X_n”

For instance, does $X_{n} / E[X_{n}]$ converge to a constant or does it have a nontrivial limiting distribution?

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Section 1, Subsection 1.3, item “Limit theorems”

Are there natural models for which $X_{n} \leq n{1-}$ with high probability for some fixed $ > 0$?

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Section 1, Subsection 1.3, item “General tree growth models”

How large is the largest common subtree of $k \geq 3$ independent UA trees with $n$ vertices each?

— On the largest common subtree of uniform attachment trees  (2609.30098 - Bäumler et al., 24 Sep 2026) in Section 1, Subsection 1.3, item “Three or more trees”