  The incidence category $\C_{\F}$ of a family $\F$ of colored posets closed
under disjoint unions and the operation of taking convex sub-posets was
introduced by the author in \cite{Sz}, where the Ringel-Hall algebra $\H_{\F}$
of $\C_{\F}$ was also defined. We show that if the Hasse diagrams underlying
$\F$ are rooted trees, then the subspace $\n_{\F}$ of primitive elements of
$\H_{\F}$ carries a pre-Lie structure, defined over $\mathbb{Z}$, and with
positive structure constants. We give several examples of $\n_{\F}$, including
the nilpotent subalgebras of $\mathfrak{sl}_n$, $L \mathfrak{gl}_n$, and
several others.
