  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.
