On the representation and enumeration of trees

Scoins(1) has shown that if Π 1 = {(1), …, ( n )} and Π 2 = {( n + 1), …, ( n + m )} are two sets of points, there are exactly m n−1 n m−1 trees of alternate parity connecting the points of Π 1 ∪ Π 2 , where each tree consists of n + m − 1 segments and each segment joins a point of Π 1 to a point of Π 2 . Another proof based on the three following results is given here.