Some refined enumerations of hybrid binary trees

hybrid binary tree is a complete binary tree where each internal node is labeled with 1 or 2, but with no left (1, 1)-edges. In this paper, we consider enumeration of the set of hybrid binary trees according to the number of internal nodes and some other combinatorial parameters. We present enumerative results by giving Riordan arrays, bivariate generating functions, as well as closed formulas. As a consequence, we obtain some new combinatorial matrices, one of which is analogous to the Borel triangle. We also present a bijection between the set of all hybrid binary trees with n internal nodes and the set of generalized Schröder paths from (0, 0) to (2 n , 0) which are consist of up steps u=(1,1) , horizontal steps h=(2,0) , down steps d=(1,-1) , and double up steps U =(2,2) , and never travel below the x -axis.