SUMMARY
Billey et al. [arXiv:1507.04976] have recently discovered a surprisingly simple formula for the number an(s)a_n(\sigma) of leaf-labelled rooted non-embedded binary trees (also known as phylogenetic trees) with n=1n\geq 1 leaves, fixed (for the relabelling action) by a given permutation s?Sn\sigma\in\frak{S}_n. Denoting by ??n\lambda\vdash n the integer partition giving the sizes of the cycles of s\sigma in non-increasing order, they show by a guessing/checking approach that if ?\lambda is a binary partition (it is known that an(s)=0a_n(\sigma)=0 otherwise), thenan(s)=l(?)?i=2(2(?i+?+?l(?))-1), a_n(\sigma)=\prod_{i=2}^{\ell(\lambda)}(2(\lambda_i+\cdots+\lambda_{\ell(\lambda)})-1), and they derive from it a formula and random generation procedure for tanglegrams (and more generally for tangled chains). Our main result is a combinatorial proof of the formula for an(s)a_n(\sigma), which yields a simplification of the random sampler for tangled chains.