Words on two letters, or their equivalent representation by ɑ sequences, label the branches of the inverse graph of the nth iterate of the parabolic map pζ (x)=ζx(2-x) of the real line. The abstract properties of words control the evolution of this graph in the content parameter. In particular, properties of words (ɑ sequences) control the process of creation and bifurcation of fixed points. The subset of lexical words of length n-1 or the corresponding set of lexical ɑ sequences of degree D=n-1 are key entities in this description, as are the divisor set of lexical words of degree D such that 1+D divides n. The parity, even or odd, of the length of the lexical sequences in the divisor set controls the motion, from left to right or right to left, of the central point (1, x(ζ)) of the inverse graph through the midpoint (1, 1), as the content parameter increase. It is proved in this paper that adjacent sequences in the ordered divisor set alternate in the parity of their lengths, this then corresponding to an oscillatory motion of the central point back and forth through the central point. The abstract parity property of words thus corresponds to an important property of the inverse graph in its evolution in the content parameter.

Trees are combinatorial structures that arise naturally in diverse applications. They occur in branching decision structures, taxonomy, computer languages, combina-torial optimization, parsing of sentences, and cluster expan-sions of statistical mechanics. Intuitively, a tree is a collection of branches connected at nodes. Formally, it can be defined as a connected graph without cycles. Schroder trees, introduced in this paper, are a class of trees for which the set of subtrees at any vertex is endowed with the structure of ordered parti-tions. An ordered partition is a partition of a set in which the blocks are linearly ordered. Labeled rooted trees and labeled planed trees are both special classes of Schroder trees. The main result gives a bijection between Schroder trees and forests of small trees-namely, rooted trees of height one. Using this bijection, it is easy to encode a Schroder tree by a sequence of integers. Several classical algorithms for trees, including a combinatorial proof of the Lagrange inversion formula, are immediate consequences of this bijection.

We present an involution for a classical identity on the alternate sum of the Gauss coefficients in terms of the traditional Ferrers diagram. It turns out that the refinement of our involution with restrictions on the height of Ferrers diagram implies a generalization of the Gauss identity, which is a terminating form of the q-Kummer identity. Furthermore, we extend the Gauss identity to the pth root of unity.

In this paper, we introduce the concepts of a formal function over an alphabet and a formal derivative based on a set of substituion rules. We call such a set of rules a context-free grammar because these rules act like a context-free grammar in the sense of a formal language. Given a context-free grammar, we can associate each formal function with an exponential formal power series. In this way, we obtain grammatical interpretations of addition, multiplication and functional composition of formal power series. A surprising fact about the grammatical calculus is that the composition of two formal power series enjoys a very simple grammatical representation. We apply this method to obtain simple demonstrations of Faa di Bruno's formula, and some identities concerning Bell polynomials, Stirling numbers and symmetric functions. In particular, the Lagrange inversion formula has a simple grammatical representation. From this point of view, one sees that Cayley's formula on labeled trees is equivalent to the Lagrange inversion formula.

We obtain an interpolation formula for symmetric functions and applications to some identities on symmetric functions, including the one obtained by Gustafson and Milne on Schur functions.