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NonRootInternalNodes on OrderedTree ​

The number of vertices that are neither the root nor a leaf.

On OrderedTree

Called as CombinatorialStat(x, "NonRootInternalNodes"). The head's page has its examples.

Live

Distribution

Over OrderedTrees at each size, tallied here in your browser; sizes past 6 are not offered.

FindStat St000168

The number of internal nodes of an ordered tree.

ObjectValue
[[]]0
[[[]]]1
[[],[]]0
[[[[]]]]2
[[[],[]]]1
[[[]],[]]1
[[],[[]]]1
[[[[[]]]]]3

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000168 has the rest.