Recognizable Formal Power Series on Trees

Trees are a very basic object in computer science. They intervene in nearly a;ly domain, and they are studied for their own, or used to represent conveniently a given situation. There are at least three directions where investigations on trees themselves are motivated, and this for different reasons. First, the notion of tree is the basis of algebraic semantics (Nivat [19], Rosen [2*2], etc.). In this context, the study of special languages of trees (i.e. forests), their classification, and their behaviour under various types of transformations are of great importance (Arnold fl], Dauchet [8 J, L,ilin 1161, Mongy [18]). By essence, work in this area is an extension of the algebraic theory of languages; trees and ianguages are in fact directly related via the derivation trees of an algebraic grammar (Thatcher [25]). A second topic heavily related to trees concerns dzra structures, Trees, mainly binary trees &and its variants, constitute one of the most widely known data structures (see e.g. Hlnuth [15]). The analysis of the worst-caLe, expected or average running time t)ehzvGrr nfi’ certain algorithms requires sometimes long and delicate computations (Flajolet [lo], Kemp [14], Flajolet and Steyaert [12]). Finally, trees occupy a distinguished place in the enumeration of graphs and maps, both because of the simplicity of their structure and for the relationship between their el-lcodings and aigebraic languages. The nature of the enumerating series, and especially the question whether they are algebraic or not, is one of the central problems in this domain (Cori [7], Chottin [4]). We propose here a theory of formal power series on trees, and present some of their basic properties together with various examples of applications which, as we hope, will show the interest of its development within the framework we just sketched. A formal power series on trees is a function which Gssociates a number to esch tree. Thus we could also have called them ‘tree functions II in analogy with the term ‘word function’ used by several authors (Paz and Salomaa [ZO], Cobham [5]) as an equivalent denomination for formal power series on words. The main goal of a formal power series is to count, or to represent the result of some computation on