Geodesic
Enumeration of weighted plane trees
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 135-144

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In weighted trees, all edges are endowed with positive integral weights. We enumerate weighted bicolored plane trees according to their weight and number of edges. 
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A. K. Zvonkin. Enumeration of weighted plane trees. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 135-144. http://geodesic.mathdoc.fr/item/FPM_2013_18_6_a7/