Non-Crossing Tree partitions are newer mathematical objects that have recent applications in genetics and mathematical biology. We explore several interesting connections between these partitions and the more commonly studied non-crossing set partitions. While non-crossing set partitions are counted by the Catalan numbers, we prove that non-crossing tree partitions in Binary trees are counted b...

It is shown that for any set of 2n points in general position in the plane there is a non-crossing perfect matching of n straight line segments whose total length is at least 2/π of the maximum possible total length of a (possibly crossing) perfect matching on these points. The constant 2/π is best possible and a non-crossing matching whose length is at least as above can be found in polynomial...