On enumerating algorithms of novel multiple leaf-distance granular regular α-subtrees of trees

Subtrees and BC-subtrees (subtrees in which the distance between any two leaves is even) are important concepts in the study of complex graphical structures. In this article, we propose a novel generalization called the leaf-distance granular regular α-tree (abbreviated as LDR α-tree for short). This is a tree in which the distance between any two leaves is divisible by α (α is a positive integer). A LDR α-subtree is simply a subtree that is also a LDR α-tree. We present basic properties and generating functions related to the LDR α-subtrees enumeration. Based on those theoretical results we provide efficient algorithms for enumerating various LDR α-subtrees of trees. Our algorithms can serve as multi-distance granularity sifters of a graph to screen all the α-subtree uniformly, and thus provide novel insights into exploring new structural properties from the perspective of multiple leaf-distance granularity.