  We calculate the superconducting critical temperature $T_c$, the singlet pair
function $\Psi^+(x)$, and triplet pair function $\Psi^-(x)$ of
superconductor/normal metal/ferromagnet (S/N/F) trilayers using the linearized
Usadel equation near $T_c$. The Green's function method developed by Fominov
$et al.$ for the S/F bilayers is extended to the S/N/F trilayer systems. The S
of the trilayers is taken to be an s-wave singlet pairing superconductor, and
the S/N and N/F interfaces are modeled in terms of the interface resistances
parameterized, respectively, by $\gamma_b^{SN}$ and $\gamma_b^{NF}$. We present
the $T_c$, $\Psi^+(x)$, and $\Psi^-(x)$ for typical $\gamma_b^{SN}$,
$\gamma_b^{NF}$, and the exchange energy $ E_{ex}$: (a) For a small (large)
$\gamma_b^{NF}$, $T_c$ of S/N/F trilayers, as $d_N$ is increased, increases
(decreases) on the length scale of N coherence length $\xi_N$ with a
discontinuity at $d_N=0$ due to a boundary condition mismatch. (b) $T_c(d_F)$
shows a non-monotonic behavior like S/F bilayers with a weakened shallow dip.
(c) The odd frequency triplet component $\Psi^-(x)$, induced by $E_{ex}$ and
proximity effects, has a maximum near the N/F interface and decreases on the
length scale $\xi_{ex}$ in F. It also penetrates into N and S regions on the
length scale $\xi_N$ and $\xi_S$, respectively. Based on these results we make
comments on the experimental observation of the odd triplet components and the
recent $T_c$ measurements in Nb/Au/CoFe trilayer systems.
