Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers.

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Counting pattern avoiding ballot paths begins with a careful analysis of the pattern. Not the length, but the characteristics of the pattern are responsible for the difficulties in finding explicit solutions. Certain features, like overlap and difference in number of → and ↑ steps determine the solution of the recursion formula. If the recursion can be solved by a polynomial sequence, we apply the Finite Operator Calculus to find an explicit form of the solution in terms of binomial coefficients.

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The purpose of this work is to extend the theory of finite operator calculus to the multivariate setting, and apply it to the enumeration of certain lattice paths. The lattice paths we consider are ballot paths. A ballot path is a path that stays weakly above the diagonal y D x, starts at the origin, and takes steps from the set ¹"; !º D ¹u; rº. Given a string p from the set ¹u; rº , we want to count the ballot paths with a given number of occurrences of p. In order to use finite operator calculus, we must put some restrictions on the string p we wish to keep track of. A ballot path ending on the diagonal can be viewed as a Dyck path, thus all of our results also apply to the enumeration of Dyck paths with a given number of occurrences of p. Finally, we give an example of counting ballot paths with a given number of occurrences of two patterns.