Enumeration of plane trees by branches and endpoints

In the count of branches, a branchpoint is defined as a point of degree at least three, and a branch is a line, or several lines in tandem, joining two branch-points, two endpoints, or a branchpoint and an endpoint. More briefly, a branch is a line in the series-reduced tree. The enumerator of plane trees by number of branches, b n (x) , is found to be b n ( x ) = ∑ 0 n − 1 ( n − 1 k ) m k x k + 1 with m 0 =∑( 2 j k c j , c n =(2 n )!/ n !( n +1)! ( m is for Th. Motzkin). The corresponding enumerator by number of endpoints other than the root, e n (x) , is shown to be equal to dn(x) , the enumerator of ballot paths by length of horizontal segments: d n ( x ) = 1 n + 1 ∑ 1 n ( n + 1 k ) ( n − 1 k − 1 ) x k , n = 1 , 2.... The enumerator by endpoints including the root, f n (x) , is shown to be f 1 ( x )= x 2 , f n (x) = e n (x) +( x −1) e n −1 ( x ), n =2,3,…. Robert Donaghey's discovery of the first result was the occasion for this paper, which presents all results in a single setting.