  Using topological string techniques, we compute BPS counting functions of 5d
gauge theories which descend from 6d superconformal field theories upon circle
compactification. Such theories are naturally organized in terms of nodes of
Higgsing trees. We demonstrate that the specialization of the partition
function as we move from the crown to the root of a tree is determined by
homomorphisms between rings of Weyl invariant Jacobi forms. Our computations
are made feasible by the fact that symmetry enhancements of the gauge theory
which are manifest on the massless spectrum are inherited by the entire tower
of BPS particles. In some cases, these symmetry enhancements have a nice
relation to the 1-form symmetry of the associated gauge theory.
