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This paper provides the generating series for the embedding of tree-likegraphs of arbitrary number of vertices, accourding to their genus. It appliesand extends the techniques of Chan, where it was used to give an alternateproof of the Goulden and Slofstra formula. Furthermore, this greatlygeneralizes the famous Harer-Zagier formula, which computes the Eulercharacteristic of the moduli space of curves, and is equivalent to thecomputation of one vertex maps.

Enumeration of Tree-like Maps with Arbitrary Number of Vertices