  For an orientation $H$ with $n$ vertices, let $T(H)$ denote the maximum
possible number of labeled copies of $H$ in an $n$-vertex tournament. It is
easily seen that $T(H) \ge n!/2^{e(H)}$ as the latter is the expected number of
such copies in a random tournament. For $n$ odd, let $R(H)$ denote the maximum
possible number of labeled copies of $H$ in an $n$-vertex regular tournament.
Adler et al. proved that, in fact, for $H=C_n$ the directed Hamilton cycle,
$T(C_n) \ge (e-o(1))n!/2^{n}$ and it was observed by Alon that already $R(C_n)
\ge (e-o(1))n!/2^{n}$. Similar results hold for the directed Hamilton path
$P_n$. In other words, for the Hamilton path and cycle, the lower bound derived
from the expectation argument can be improved by a constant factor. In this
paper we significantly extend these results and prove that they hold for a
larger family of orientations $H$ which includes all bounded degree Eulerian
orientations and all bounded degree balanced orientations, as well as many
others. One corollary of our method is that for any $k$-regular orientation $H$
with $n$ vertices, $T(H) \ge (e^k-o(1))n!/2^{e(H)}$ and in fact, for $n$ odd,
$R(H) \ge (e^k-o(1))n!/2^{e(H)}$.
