The number of crossing-similar perfect matchings of a perfect matching. Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$. Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of crossings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$. [thm 1.2, 1] shows that to find out whether $M$ and $N$ are crossing-similar, it is enough to check that $M$ and $N$ have the same number of crossings, and that the distribution of crossings agrees for their direct children. [thm 3.3, 1], see also [2], gives the number of equivalence classes of crossing-similar matchings with $n$ arcs as $$2^{n-2}\left(\binom{n}{2}+2\right).$$