A new pattern-enumeration formula for ordered trees that includes nonleaf-only pattern components is proven and used to count protected and unprotected nodes, stumps, and related tree families.
Some bijections and identities for the Catalan and Fine numbers
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Nonleaf Patterns in Trees: Protected Nodes and Fine Numbers
A new pattern-enumeration formula for ordered trees that includes nonleaf-only pattern components is proven and used to count protected and unprotected nodes, stumps, and related tree families.