Enumeration of Weighted Plane Trees

In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges. 1 Preliminaries This note is not intended for a journal publication: it does not contain difficult results, and the proofs use only standard and well-known techniques. Furthermore, some of the results are already known. However, they have their place in the context of the study of weighted trees, see [5], [6]. Definition 1.1 (Weighted tree) A weighted bicolored plane tree, or a weighted tree, or just a tree for short, is a bicolored plane tree whose edges are endowed with positive integral weights. The sum of the weights of the edges of a tree is called the total weight of the tree. The degree of a vertex is the sum of the weights of the edges incident to this vertex. Obviously, the sum of the degrees of black vertices, as well as the sum of the degrees of white vertices, is equal to the total weight n of the tree. Let the tree have p black vertices, of degrees α1, . . . , αp, and q white vertices, of degrees β1, . . . , βq, respectively. Then the pair of partitions (α, β), α, β ⊢ n, is called passport of the tree. The weight distribution of a weighted tree is a partition μ ⊢ n, μ = (μ1, μ2, . . . , μm) where m = p+ q − 1 is the number of edges, and μi, i = 1, . . . ,m are the weights of the edges. Leaving aside the weights and considering only the underlying plane tree, we speak of a topological tree, which is a bicolored plane tree. Weighted trees whose weight distribution is μ = 1 will be called ordinary trees: they coincide with the corresponding topological trees. The adjective plane in the above definition means that our trees are considered not as mere graphs but as plane maps. More precisely, this means that the cyclic order of branches around each vertex of the tree is fixed, and changing this order will in general give a different tree. All the trees considered in this paper will be endowed with the “plane” structure; therefore, the adjective “plane” will often be omitted. Example 1.2 (Weighted tree) Figure 1 shows an example of a weighted tree. The total weight of this tree is n = 18; its passport is (α, β) = (521, 7641); the weight distribution is μ = 5321. Definition 1.3 (Rooted tree) A tree with a distinguished edge is called a rooted tree, and the distinguished edge itself is called its root. We consider the root edge as being oriented from black to white. The goal of this paper is the enumeration of rooted weighted (bicolored plane) trees. ∗LaBRI, UMR 5800, Université de Bordeaux, Talence, 33400, France. E-mail: [email protected]