Nonleaf patterns in trees: Protected nodes and fine numbers

Nachum Dershowitz*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We derive a closed-form formula for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also contain components that match subtrees or sequences of subtrees. This result extends previous versatile tree-pattern enumeration formulæ to incorporate components that are only allowed to match nonleaf subtrees and provides enumerations of trees by the number of protected (shortest outgoing path has two or more edges) or unprotected nodes.

Original languageEnglish
Article number20.7.7
Pages (from-to)1-17
Number of pages17
JournalJournal of Integer Sequences
Issue number7
StatePublished - 2020


  • Dyck path
  • Fine number
  • Fine path
  • Hill
  • Lattice path
  • Protected node
  • Tree enumeration
  • Tree pattern


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