How many times can two minimum spanning trees cross?

Let $P$ be a generic set of $n$ points in the plane, and let $P=R\cup B$ be a coloring of $P$ in two colors. We are interested in the number of crossings between the minimum spanning trees (MSTs) of $R$ and $B$, denoted by $\text{crossAB}(R,B)$. We define the \emph{bicolored MST crossing number} of $P$, denoted by $\text{cross}(P)$, as $\text{cross}(P)={\max }_{P=R\cup B}(\text{crossAB}(R,B))$. We prove a linear upper bound for $\text{cross}(P)$ when $P$ is generic. If $P$ is dense or in convex position, we provide linear lower bounds. Lastly, if $P$ is chosen uniformly at random from the unit square and is colored uniformly at random, we prove that the expected value of $\text{crossAB}(R,B)$ is linear.