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Oct 3, 2015 at 0:54 comment added Tom Copeland Must somehow be related to "Expansions that grow on trees" by Iserles (ams.org/notices/200204/fea-iserles.pdf) and "The Magnus expansion, trees, and the Knuth correspondence" by Ebrahimi-Fard and Manchon (arxiv.org/abs/1203.2878) ? Your trees must be enumerated on the OEIS. If not,please add an entry there for other researchers.
Sep 2, 2015 at 4:13 history answered Theo Johnson-Freyd CC BY-SA 3.0