Subtrees and BC-subtrees of maximum degree no more than k in trees

The subtrees and BC-subtrees (subtrees where any two leaves are at even distance apart) have been extensively studied in recent years. Such structures, under special constraints on degrees, have applications in many fields. Through an approach based on generating functions, we present recursive algorithms for enumerating various subtrees and BC-subtrees of maximum degree $\leq k$ in trees. The algorithms are illustrated through detailed examples. We also briefly discuss, in trees, the densities of subtrees (resp.~BC-subtrees) of maximum degree $\leq k(\geq 2)$ among all subtrees (resp.~BC-subtrees).