  We prove that the Catalan Lie idempotent $D_n(a,b)$, introduced in
  [Menous {\it et al.}, Adv. Appl. Math. 51 (2013), 177] can be
  refined by introducing $n$ independent parameters $a_0,\ldots,a_{n-1}$
  and that the coefficient of each monomial is itself a Lie idempotent
  in the descent algebra. These new idempotents are multiplicity-free
  sums of subsets of the Poincar\'e-Birkhoff-Witt basis of the Lie module.
  These results are obtained by embedding noncommutative symmetric functions
into the
  dual noncommutative Connes-Kreimer algebra, which also allows us to
interpret, and rederive in a simpler way,
  Chapoton's results on a two-parameter tree expanded series.
