BinaryTree from OrderedTree

An ordered tree as a binary tree with one fewer node: a vertex's left brother becomes its left child, its last child its right child, and the root goes.

OrderedTree → BinaryTreeinverse

FindStat's Mp00049, Sage's to_binary_tree_left_branch, the inverse of OrderedTree(BinaryTree). RightBranchBinaryTree is the same with the roles of left and right exchanged.

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

Live

Explore

Each element of OrderedTree at a size, and its image, applied in your browser; sizes past 6 are not offered.

Size

Properties

Sizes 1 to

Transport a statistic

Pick a statistic on BinaryTree. Its distribution over the images of OrderedTree at each size is computed, and compared with the 21 statistics on OrderedTree: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic HeightLeftSubtreeSizeLeftSpineLeavesUnaryNodesBinaryNodesStrahlerNumberPathLengthExternalPathLengthPitchforksTreeFactorialLinearExtensionCountCollessIndexMaximalLeftBranchesDistinctSubtreesProtectionNumberRightChildCountRightSubtreeSizeSumParenthesisCodeValueBoundaryNodesNonBoundaryNodesNonRootBinaryNodesHookNumberVertexHeightSumRightRightPathCountRightRightRightPathCountRightRightRightRightPathCountRightRightLeftPathCountRightRightRightLeftPathCountRightLeftLeftPathCountRightLeftLeftLeftPathCountRightRightLeftLeftPathCountRightRightLeftRightPathCountRightForkRightLeftPatternCountForkRightRightLeftPatternCountSwapEquivalentTreeCountSwapEquivalentTreeCountLog Sizes 1 to

Also from

The same name out of other carriers, each its own implementation: BinaryTreeParentArray, DyckPath.

FindStat Mp00049

to binary tree: left brother = left child

Matched by value: our map agrees with FindStat's image on each of the 197 rows of size at most 6 we can read (0 unread; its table has 1200 rows).

BijectionGraded mapas FindStat records them, checked by FindStat's contributors.

Return a binary tree of size $n-1$ (where $n$ is the size of $t$, and where $t$ is an ordered tree) by the following recursive rule:

Object	Image
[]	.
[[]]	[.,.]
[[[]]]	[.,[.,.]]
[[],[]]	[[.,.],.]

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00049 has the rest.