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.
$k$-protected vertices in unlabeled rooted plane trees
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abstract
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size $n$ approaches 0.727649, and the average rank of the root of a tree of size $n$ approaches 1.62297.
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2019 1verdicts
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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.