  Let $F=\{H_1,...,H_k\}$ be a family of graphs. A graph $G$ with $m$ edges is
called {\em totally $F$-decomposable} if for {\em every} linear combination of
the form $\alpha_1 e(H_1) + ... + \alpha_k e(H_k) = m$ where each $\alpha_i$ is
a nonnegative integer, there is a coloring of the edges of $G$ with
$\alpha_1+...+\alpha_k$ colors such that exactly $\alpha_i$ color classes
induce each a copy of $H_i$, for $i=1,...,k$. We prove that if $F$ is any fixed
family of trees then $\log n/n$ is a sharp threshold function for the property
that the random graph $G(n,p)$ is totally $F$-decomposable. In particular, if
$H$ is a tree, then $\log n/n$ is a sharp threshold function for the property
that $G(n,p)$ contains $\lfloor e(G)/e(H) \rfloor$ edge-disjoint copies of $H$.
