  Linked cluster expansions are generalized from an infinite to a finite volume
on a $d$-dimensional hypercubic lattice. They are performed to 20th order in
the expansion parameter to investigate the phase structure of scalar $O(N)$
models for the cases of $N=1$ and $N=4$ in 3 dimensions. In particular we
propose a new criterion to distinguish first from second order transitions via
the volume dependence of response functions for couplings close to but not at
the critical value. The criterion is applicable to Monte Carlo simulations as
well. Here it is used to localize the tricritical line in a $\Phi^4 + \Phi^6$
theory. We indicate further applications to the electroweak transition.
