Counting glycans revisited

We present an algorithm for counting glycan topologies of order n that improves on previously described algorithms by a factor n in both time and space. More generally, we provide such an algorithm for counting rooted or unrooted d-ary trees with labels or masses assigned to the vertices, and we give a "recipe" to estimate the asymptotic growth of the resulting sequences. We provide constants for the asymptotic growth of d-ary trees and labeled quaternary trees (glycan topologies). Finally, we show how a classical result from enumeration theory can be used to count glycan structures where edges are labeled by bond types. Our method also improves time bounds for counting alkanes.