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For similar recurrences (and a great reference on all sorts of results on counting labeled trees), see J. W. Moon's Counting Labelled Trees, especially section 3.8, math.ucla.edu/~pak/hidden/papers/….
See Concrete Mathematics (2nd ed.) by Graham, Knuth, and Patashnik, formula (7.51), page 351, and formula (7.58), page 352. (They have a slightly different definition of Stirling polynomials.)
It should be pointed out that this theorem was found earlier by Ralph Fröberg, Determination of a class of Poincaré series. Mathematica Scandinavica, 37(1) 29–39, 1975 and L. Carlitz, R. Scoville, and T. Vaughan, Enumeration of pairs of sequences by rises, falls and levels. Manuscripta Mathematica, 19, 211–243, 1976.
A combinatorial approach to Lie series has been given by Gilbert Labelle in two papers: MR0814421 (87c:05007) Une combinatoire sous-jacente au théorème des fonctions implicites. [A combinatorial theory underlying the implicit function theorem] J. Combin. Theory Ser. A 40 (1985), no. 2, 377–393 and MR0787718 (86j:05015) Éclosions combinatoires appliquées à l'inversion multidimensionnelle des séries formelles. [Combinatorial bloomings applied to the multidimensional inversion of formal series] J. Combin. Theory Ser. A 39 (1985), no. 1, 52–82.
