For a probability distribution $(p_s, s \in S)$ on a finite set $S$, call a random forest $F$ of rooted trees labeled by $S$ (with edges directed away from the roots) a {\em $p$-forest} if given $F$ has $m$ edges the vector of out-degrees of vertices of $F$ has a multinomial distribution with parameters $m$ and $(p_s,s \in S)$, and given also these out-degrees the distribution of $F$ is uniform on all forests with the given out-degrees. The family of distributions of $p$-forests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries.