The likely maximum size of twin subtrees in a large random tree

We call a pair of vertex-disjoint, induced subtrees of a rooted trees twins if they have the same counts of vertices by out-degrees. The likely maximum size of twins in a uniformly random, rooted Cayley tree of size $n\to\infty$ is studied. It is shown that the expected number of twins of size $(2+\delta)\sqrt{\log n\cdot\log\log n}$ approaches zero, while the expected number of twins of size $(2-\delta)\sqrt{\log n\cdot\log\log n}$ approaches infinity.