For a labeled tree $ T $ sure $ {1,2, …, n } $ a *inversion* of $ T $ is a pair $ 1 <i <j leq n $ such as $ j belongs to the unique path of $ 1 $ at $ i $ (we think of $ T $ as being rooted in $ 1 $). Let $ mathrm {inv} (T) $ indicate the number of inversions of $ T $.

Define the generating function $ f (q): = sum_ {T} q ^ { mathrm {inv} (T)} $ where the sum is on all the trees labeled on $ {1,2, …, n } $.

So we know that $ f (-1) $ is the number of alternate permutations in $ mathfrak {S} _n $ (that is to say the so-called "Euler number"). See, for example, Goulden-Jackson 3.3.49 (d).

**Question**: Is there a simple proof of this result via inversion inversion of sign?