  For a complete graph $K_n$ of order $n$, an edge-labeling $c:E(K_n)\to \{
-1,1\}$ satisfying $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$, we
consider the problem to minimize $|c(E(F'))|$ over all isomorphic copies $F'$
of $F$ in $K_n$. In particular, we ask under which additional conditions there
is a zero-sum copy, that is, a copy $F'$ of $F$ with $c(E(F'))=0$.
  We show that there is always a copy $F'$ of $F$ with $|c(E(F'))|\leq
\Delta(F)+1$, where $\Delta(F)$ is the maximum degree of $F$. We conjecture
that this bound can be improved to $|c(E(F'))|\leq (\Delta(F)-1)/2$ and verify
this for $F$ being the star $K_{1,n-1}$. Under some simple necessary
divisibility conditions, we show the existence of a zero-sum $P_3$-factor, and,
for sufficiently large $n$, also of a zero-sum $P_4$-factor.
