Geodesic
The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods
Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 12-19

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A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements l ij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions. 
Jackson, D. M.; Goulden, I. P. The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods. Canadian journal of mathematics, Tome 33 (1981) no. 1, pp. 12-19. doi: 10.4153/CJM-1981-002-2
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