  Many interesting problems are obtained by attempting to generalize classical
results on convexity in Euclidean spaces to other convexity spaces, in
particular to convexity spaces on graphs. In this paper we consider
$P_3$-convexity on graphs. A set $U$ of vertices in a graph $G$ is $P_3$-convex
if every vertex not in $U$ has at most one neighbour in $U$. More specifically,
we consider Radon numbers for $P_3$-convexity in trees.
  Tverberg's theorem states that every set of $(k-1)(d+1)-1$ points in
$\mathbb{R}^d$ can be partitioned into $k$ sets with intersecting convex hulls.
As a special case of Eckhoff's conjecture, we show that a similar result holds
for $P_3$-convexity in trees.
  A set $U$ of vertices in a graph $G$ is called free, if no vertex of $G$ has
more than one neighbour in $U$. We prove an inequality relating the Radon
number for $P_3$-convexity in trees with the size of a maximal free set.
