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RightLeftLeftLeftPathCount on BinaryTree ​

The number of nodes from which taking the right, then the left, then the left, then the left child stays inside the tree.

On BinaryTree

A contiguous pattern of nodes along one path, the subtrees off it anything.

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

Live

Distribution

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

FindStat St000131

The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],.

The number of occurrences of the contiguous pattern {{{[.,[[[[.,.],.],.],.]]}}} in a binary tree.
ObjectValue
[.,.]0
[.,[.,.]]0
[[.,.],.]0
[.,[.,[.,.]]]0
[.,[[.,.],.]]0
[[.,.],[.,.]]0
[[.,[.,.]],.]0
[[[.,.],.],.]0

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000131 has the rest.