An Improved Lower Bound on the Largest Common Subtree of Random Leaf-Labeled Binary Trees

... On the other hand, a polynomial lower bound of order n 1/8 was obtained by Bernstein, Ho, Long, Steel, St. John and Sullivant in [7]. This lower bound was recently improved to n ( √ 3−1)/2 ≈ n 0.366 by Aldous [5] and to n 0.4464 by Khezeli [19] in expectation. Finally, we also mention that √ n has been proved to be the right order of magnitude if the trees T n and T ′ n are conditioned to have the same shape [25], and that the upper bound in √ n holds robustly for many random trees models arising from branching processes [27]. ...

... This compact, continuous random tree with fractal dimension 2 was introduced in [2] and can be built in a natural way from a normalized Brownian excursion (see Section 2.2 for complete definitions). It also has the important property that its branching points all have degree 3. We highlight that comparisons between the discrete trees T n and the continuous object T already play an important role in the proofs of the lower bounds of [5] and [19]. ...

... Indeed, as can be seen on Figure 1, a common subtree of two trees T n and T ′ n gives a "correspondence" between a part of T n and a part of T ′ n , which can be extended to a homeomorphism in the continuous limit. This is not a completely new idea, as the arguments of [5] (and the improvements done in [19]) can already be interpreted as a proof of the existence of a homeomorphism from T to T ′ with a certain Hölder exponent. As we check in Theorem 22 in the appendix, the actual Hölder exponent given by [5] turns out to be 5 − 2 √ 6 ≈ 0.1010. ...