Question 122
A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $A$ and $B$. An $\mathcal{A}\mathcal{B}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $A$ and the other in $B$, and such that no segments form a closed polyline. An $\mathcal{A}\mathcal{B}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$ and $A_2B_2$ in the $\mathcal{A}\mathcal{B}$-tree such that $A_1$ is in $A$ and $A_1B_1 + A_2B_2 \gt A_1B_2 + A_2B_1$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{A}\mathcal{B}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.