OrderedTree from BinaryTree

A binary tree as an ordered tree with one more vertex: a left child becomes its parent's left brother, a right child its parent's last child.

BinaryTree → OrderedTreeinverse

FindStat's Mp00010, Sage's to_ordered_tree_left_branch, a bijection from the binary trees with n nodes to the ordered trees with n edges. The inverse is BinaryTree(OrderedTree); RightBranchOrderedTree is the same with the roles of left and right exchanged.

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

Live

Explore

Each element of BinaryTree 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 OrderedTree. Its distribution over the images of BinaryTree at each size is computed, and compared with the 37 statistics on BinaryTree: those that share it are the ones the map sends it to, as far as the sizes computed show.

Statistic RootDegreeHeightLeavesNonRootInternalNodesLargestOutDegreeStrahlerNumberPathLengthTreeFactorialLinearExtensionCountDistinctSubtreesProtectionNumberBoundaryLengthTrunkLengthBreadthLongestPathVertexCountAutomorphismGroupSizeInequivalentIncreasingTreeCountOneProtectedNodeCountTwoProtectedNodeCountPruningNumberBoundaryLengthBeyondTrunk Sizes 1 to

Also from

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

FindStat Mp00010

to ordered tree: left child = left brother

Matched by value: our map agrees with FindStat's image on each of the 196 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 an ordered tree of size $n+1$ by the following recursive rule:

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/Mp00010 has the rest.