The vanishing levels of a tree

Abstract We initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa $ -tree $\mathbf T$ . This is an invariant in the sense that if $\mathbf T$ and $\mathbf T'$ are club-isomorphic, then $V(\mathbf T)\mathbin {\bigtriangleup } V(\mathbf T')$ is nonstationary. Additional features of this invariant imply that the spectrum is closed under finite unions and intersections. The set $V(\mathbf T)$ must be stationary for a homogeneous normal $\kappa $ -Aronszajn tree $\mathbf T$ , and if there exists a special $\kappa $ -Aronszajn tree, then there exists one $\mathbf T$ that is homogeneous and satisfies that $V(\mathbf T)$ covers a club in $\kappa $ . It is consistent (from large cardinals) that there is an $\aleph _2$ -Souslin tree, and yet $V(\mathbf T)$ is co-stationary for every $\aleph _2$ -tree $\mathbf T$ . Both $V(\mathbf T)=\emptyset $ and $V(\mathbf T)=\kappa $ (modulo nonstationary) are shown to be feasible using $\kappa $ -Souslin trees, even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^\kappa $ many $\kappa $ -Souslin trees for which the corresponding family of vanishing levels forms an antichain in the Boolean algebra of the powerset of $\kappa $ , modulo the nonstationary ideal.