Sequences counting series-reduced and lone-child-avoiding trees by number of vertices. By Gus Wiseman 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. Latest version: https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub