The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods

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. Remarkably, the number of such trees which are planted is a multiple of the number of spanning arbores­ cences on a graph with the edge partition as its adjacency matrix. This result is used (Section 4) to obtain the number of rooted plane i£-chromatic trees with fixed chromatic partition, a number given by Tutte [5] for planted trees in the case K = 2. Finally, a new combinatorial correspon­ dence between two sets of trees is given (Section 5) which yields the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte (BEST) theorem as a special case. We use a familiar decomposition of rooted trees. Associated with this is a system of functional equations which may be solved by a specialisa­ tion, given in Section 2, of the Lagrange theorem in many variables. The specialisation accounts for the persistence, in combinatorial enumeration, of determinants of matrices with row and column constraints (see, for example, [6]). Throughout this paper we use the notation: the number of non-root vertices of colour i is nt = ]C^i/