Largest common subcladogram exponents

Definitions and mathematical statement

A rooted binary cladogram on $[n]=\{1,\ldots,n\}$ is a finite rooted tree whose leaves carry these distinct labels, whose internal vertices have two unordered children, and whose internal vertices carry no labels. For $A\subseteq[n]$, its induced cladogram retains the ancestry connecting the leaves of $A$, takes their most recent common ancestor as root, and suppresses vertices with only one retained child.

Consider separately two probability laws: the uniform law on the $(2n-3)!!$ such cladograms, and the coalescent law obtained by starting with $n$ single-leaf lineages and repeatedly choosing uniformly a pair of current lineages and joining them under a new parent.

Under either law, sample $T_1,T_2$ independently and set \[ C_n=\max\{|A|:A\subseteq[n],\ T_1|_A\cong T_2|_A\}, \] where the isomorphism must preserve the root and each leaf label. Prove that there are constants $\gamma_a,\gamma_b<1/2$, respectively for the uniform and coalescent laws, for which \[ \mathbb EC_n=n^{\gamma_a+o(1)},\qquad \mathbb EC_n=n^{\gamma_b+o(1)}, \] and identify or characterize these exponents.

The statements are equivalent to convergence of $\log\mathbb EC_n/\log n$ to the corresponding constants. The same subset of labels must be used in both trees; arbitrary relabeling is not permitted. The requested result concerns the optimum common induced cladogram, not merely the output of one particular recursive matching algorithm.

Short English statement

What are the growth exponents of the largest common labeled subcladogram in two independent uniform or coalescent random trees?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 9700019 (AMR-096-0019), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Aldous - Open problems, substructures.html; linked 2003 document, Example 3. Original source indexed by AMR-096-0019. https://www.stat.berkeley.edu/~aldous/Research/OP/index.html.
• S3 D. Aldous, Largest common substructures in probabilistic combinatorics, original problem definitions. https://www.stat.berkeley.edu/~aldous/Research/OP/common_subs.pdf.