Vertex topological indices and tree expressions, generalizations of continued fractions

We expand on the work of Hosoya to describe a generalization of continued fractions called "tree expressions." Each rooted tree will be shown to correspond to a unique tree expression which can be evaluated as a rational number (not necessarily in lowest terms) whose numerator is equal to the Hosoya index of the entire tree and whose denominator is equal to the tree with the root deleted. In the development, we use Z(G) to define a natural candidate ζ(G, v) for a "vertex topological index" which is a value applied to each vertex of a graph, rather than a value assigned to the graph overall. Finally, we generalize the notion of tree expression to "labeled tree expressions" that correspond to labeled trees and show that such expressions can be evaluated as quotients of determinants of matrices that resemble adjacency matrices.