Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

By Gus Wiseman

Last updated: Jan 20 2020

- We say that a rooted tree is lone-child-avoiding (LCA) if no vertex has exactly one child. The Matula-Goebel numbers of these trees are given by A291636.

- We say that a (not necessarily rooted) tree is topologically series-reduced (TSR) if no vertex (including the root) has degree 2. The Matula-Goebel numbers of these trees (in the rooted case) are given by A331489.

- These two concepts are used in ambiguous, confusing, or erroneous ways in many OEIS entries (present author not excepted).

Unlabeled rooted trees:

  LCA: A001678 (shifted left once)

  TSR: A001679

Labeled rooted trees:

  LCA: A060356

  TSR: A060313

Unlabeled unrooted trees:

  LCA: not well-defined

  TSR: A000014

Labeled unrooted trees:

  LCA: A108919

  TSR: A005512

Note that for n > 1, we have A331488(n) = A001679(n) - A001678(n). See also: A000311, A059123, A198518, A254382.