  We analyse the computation of the partition function of 5d $T_N$ theories in
Higgs branches using the topological vertex. The theories are realised by a web
of $(p,q)$ 5-branes whose dual description may be given by an M-theory
compactification on a certain local non-toric Calabi-Yau threefold. We
explicitly show how it is possible to directly apply the topological vertex to
the non-toric geometry. Using this novel technique, which considerably
simplifies the computation by the existing method, we are able to compute the
partition function of the higher rank $E_6$, $E_7$ and $E_8$ theories. Moreover
we show how in some specific cases similar results can be extended to the
computation of the partition function of 5d $T_N$ theories in the Higgs branch
using the refined topological vertex. These cases require a modification of the
refined topological vertex.
