Background:The frequency of small subtrees in biological,social,and other types of networks could shed light into the structure,function,and evolution of such networks.However,counting all possible subtrees of a prescribed size can be computationally expensive because of their potentially large number even in small,sparse networks.Moreover,most of the existing algorithms for subtree counting belong to the subtree-centric approaches,which search for a specific single subtree type at a time,potentially taking more time by searching again on the same network.Methods:In this paper,we propose a network-centric algorithm (MTMO) to efficiently count k-size subtrees.Our algorithm is based on the enumeration of all connected sets of k-1 edges,incorporates a labeled rooted tree data structure in the enumeration process to reduce the number of isomorphism tests required,and uses an array-based indexing scheme to simplify the subtree counting method.Results:The experiments on three representative undirected complex networks show that our algorithm is roughly an order of magnitude faster than existing subtree-centric approaches and base network-centric algorithm which does not use rooted tree,allowing for counting larger subtrees in larger networks than previously possible.We also show major differences between unicellular and multicellular organisms.In addition,our algorithm is applied to find network motifs based on patte growth approach.Conclusions:A network-centric algorithm which allows for a faster counting of non-induced subtrees is proposed.This enables us to count larger motif in larger networks than previously.