The Likely Maximum Size of Twin Subtrees in a Large Random Tree

We call a pair of vertex-disjoint, induced subtrees of a rooted tree 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 \(n\rightarrow \infty \) 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.