A288953 Number of relaxed compacted binary trees of right height at most one with minimal sequences between branch nodes except after the last branch node on level 0.

1, 1, 3, 10, 51, 280, 1995, 15120, 138075, 1330560, 14812875, 172972800, 2271359475, 31135104000, 471038042475, 7410154752000, 126906349444875, 2252687044608000, 43078308695296875, 851515702861824000, 17984171447178811875, 391697223316439040000

Offset

0,3

Comments

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. A branch node is a node with a left and right edge (no pointer). See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

a(n) is the number of plane increasing trees with n+1 nodes where in the growth process induced by the labels maximal young leaves and non-maximal young leaves alternate except for a sequence of maximal young leaves at the beginning. A young leaf is a leaf with no left sibling. A maximal young leaf is a young leaf with maximal label. See the Wallner link. - Michael Wallner, Apr 20 2017

Links

Table of n, a(n) for n=0..21.

Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017

Formula

E.g.f.: (2-z)/(3*(1-z)^2) + 1/(3*sqrt(1-z^2)).

Example

Denote by L the leaf and by o nodes. Every node has exactly two out-going edges or pointers. Internal edges are denoted by - or |. Pointers are omitted and may point to any node further right. The root is at level 0 at the very left.
The general structure is
  L-o-o-o-o-o-o-o-o
          | | | | |
          o o o o o.
For n=0 the a(0)=1 solution is L.
For n=1 the a(1)=1 solution is L-o.
For n=2 the a(2)=3 solutions are
L-o-o     L-o
            |
            o
  2    +   1    solutions of this shape with pointers.

Crossrefs

Cf. A288954 (variation with additional initial sequence).

Cf. A177145 (variation without final sequence).

Cf. A001147 (relaxed compacted binary trees of right height at most one).

Cf. A082161 (relaxed compacted binary trees of unbounded right height).

Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288954 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).

Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of relaxed compacted binary trees of right height at most one, see the Wallner link).

Sequence in context: A341648 A307099 A048175 * A192482 A233537 A020132

Adjacent sequences: A288950 A288951 A288952 * A288954 A288955 A288956

Keyword

nonn

Author

Michael Wallner, Jun 20 2017

Status

approved