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We prove a surprising symmetry between the law of the size $G_n$ of thegreedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size $n$and that of its complement. We show that $G_n$ has the same law as the numberof vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. Thisenables us to compute the exact law of the $G_n$. We also give a Markovianconstruction of the greedy independent set, which highlights the symmetry of$G_n$ and whose proof uses a new Markovian exploration of rooted Cayley treeswhich is of independent interest.

A surprising symmetry for the greedy independent set on Cayley trees