Journal of Mathematical Biology

, Volume 69, Issue 4, pp 799–816 | Cite as

Counting glycans revisited

  • Sebastian Böcker
  • Stephan Wagner


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.


Counting glycans Counting chemical structures Counting trees Pólya’s enumeration theorem Algorithms 

Mathematics Subject Classification




We thank Birte Kehr for preparing Fig. 1. This material is based upon work supported financially by the National Research Foundation of South Africa under grant number 70560.

Supplementary material

285_2013_721_MOESM1_ESM.txt (3 kb)
Supplementary material 1 (txt 2 KB)
285_2013_721_MOESM2_ESM.groovy (2 kb)
Supplementary material 2 (groovy 1 KB)


  1. Bell JP, Burris SN, Yeats KA (2006) Counting rooted trees: the universal law \(t(n)\sim C\rho ^{-n} n^{-3/2}\). Electron J Combin 13(1): R63, 64 (electronic)Google Scholar
  2. Böcker S, Kehr B, Rasche F (2011) Determination of glycan structure from tandem mass spectra. IEEE/ACM Trans Comput Biol Bioinform 8(4):976–986CrossRefGoogle Scholar
  3. Böcker S, Lipták Zs (2005) Efficient mass decomposition. In: Proccedings of the ACM symposium on applied computing (ACM SAC 2005), ACM Press, Santa Fe, pp 151–157Google Scholar
  4. Böcker S, Lipták Zs (2007) A fast and simple algorithm for the money changing problem. Algorithmica 48(4):413–432MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cayley A (1881) On the analytical forms called trees. Am J Math 4:266–268MathSciNetCrossRefzbMATHGoogle Scholar
  6. Ethier M, Saba JA, Spearman M, Krokhin O, Butler M, Ens W, Standing KG, Perreault H (2003) Application of the StrOligo algorithm for the automated structure assignment of complex N-linked glycans from glycoproteins using tandem mass spectrometry. Rapid Commun Mass Spectrom 17(24):2713–2720CrossRefGoogle Scholar
  7. Flajolet P, Sedgewick R (2009) Analytic combinatorics. Cambridge University Press. Available from
  8. Gaucher SP, Morrow J, Leary JA (2000) STAT: a saccharide topology analysis tool used in combination with tandem mass spectrometry. Anal Chem 72(11):2331–2336CrossRefGoogle Scholar
  9. Goldberg D, Bern MW, Li B, Lebrilla CB (2006) Automatic determination of O-glycan structure from fragmentation spectra. J Proteome Res 5(6):1429–1434CrossRefGoogle Scholar
  10. Harary F, Palmer EM (1973) Graphical enumeration. Academic Press, New YorkzbMATHGoogle Scholar
  11. Harary F, Robinson RW, Schwenk AJ (1975) Twenty-step algorithm for determining the asymptotic number of trees of various species. J Aust Math Soc Ser A 20(4):483–503MathSciNetCrossRefzbMATHGoogle Scholar
  12. Henze HR, Blair CM (1931) The number of structurally isomeric alcohols of the methanol series. J Am Chem Soc 53(8):3042–3046CrossRefGoogle Scholar
  13. Jayo RG, Li J, Chen DD (2013) Capillary electrophoresis mass spectrometry for the characterization of O-acetylated N-glycans from fish serum. Anal Chem 84(20):8756–8762CrossRefGoogle Scholar
  14. Otter R (1948) The number of trees. Ann Math 49(3):583–599MathSciNetCrossRefzbMATHGoogle Scholar
  15. Palmisano G, Antonacci D, Larsen MR (2010) Glycoproteomic profile in wine: a ‘sweet’ molecular renaissance. J Proteome Res 9(12):6148–6159CrossRefGoogle Scholar
  16. Pólya G (1937) Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math 68(1):145–254MathSciNetCrossRefGoogle Scholar
  17. Rains EM and Sloane NJA (1999) On Cayley’s enumeration of alkanes (or 4-valent trees). J Integer Seq 2:(electronic)Google Scholar
  18. Raman R, Raguram S, Venkataraman G, Paulson JC, Sasisekharan R (2005) Glycomics: an integrated systems approach to structure-function relationships of glycans. Nat Methods 2(11):817–824CrossRefGoogle Scholar
  19. Trinajstić N, Jeričević Ž, Knop JV, Müller WR, Szymanski K (1983) Computer generation of isomeric structures. Pure Appl Chem 55(2):379–390Google Scholar
  20. Varki A, Cummings RD, Esko JD, Freeze HH, Stanley P, Bertozzi CR, Hart GW, Etzler ME (eds) (2009) Essentials of glycobiology, 2nd edn. Cold Spring Harbor Laboratory Press. Available from

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Lehrstuhl für BioinformatikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

Personalised recommendations