Geodesic
Enumeration of one class of plane weighted trees
Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 171-184

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We present an enumeration formula for weighted trees, i.e., trees where vertices and edges have weights (a weight is a positive integer) and the weight of each vertex is equal to the sum of the weights of the edges incident to it. Each tree has a binary structure: we can color its vertices in two colors, black and white, so that adjacent vertices have different colors. In this work, the following problem is considered: enumerate weighted plane trees with given sets of weights of black and white vertices. 
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     author = {Yu. Yu. Kochetkov},
     title = {Enumeration of one class of plane weighted trees},
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Yu. Yu. Kochetkov. Enumeration of one class of plane weighted trees. Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 6, pp. 171-184. http://geodesic.mathdoc.fr/item/FPM_2013_18_6_a10/