# Seven Trees in One

- Published in 1994
- Added on

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Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the bijection to a seven-tuple of trees involves case distinctions only down to a fixed depth (namely four) in the given seven-tuple. We clarify how this and similar bijections are related to the free commutative semiring on one generator X subject to X=1+X^2. Finally, our main theorem is that the existence of particularly elementary bijections can be deduced from the provable existence, in intuitionistic type theory, of any bijections at all.

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### BibTeX entry

@article{SevenTreesinOne, title = {Seven Trees in One}, abstract = {Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the bijection to a seven-tuple of trees involves case distinctions only down to a fixed depth (namely four) in the given seven-tuple. We clarify how this and similar bijections are related to the free commutative semiring on one generator X subject to X=1+X^2. Finally, our main theorem is that the existence of particularly elementary bijections can be deduced from the provable existence, in intuitionistic type theory, of any bijections at all.}, url = {http://arxiv.org/abs/math/9405205v1 http://arxiv.org/pdf/math/9405205v1}, year = 1994, author = {Andreas Blass}, comment = {}, urldate = {2018-11-26}, archivePrefix = {arXiv}, eprint = {math/9405205}, primaryClass = {math.LO}, collections = {fun-maths-facts,unusual-arithmetic} }