A Cantor-Bendixson Rank for Siblings of Trees

Similar to topological spaces, we introduce the Cantor-Bendixson rank of a tree $T$ by repeatedly removing the leaves and the isolated vertices of $T$ using transfinite recursion. Then, we give a representation of a tree $T$ as a leafless tree $T^{\mathrm{\infty }}$ with some leafy trees attached to $T^{\mathrm{\infty }}$. With this representation at our disposal, we count the siblings of a tree and obtain partial results towards a conjecture of Bonato and Tardif.