We study the problem of statistical analysis on sets of phylogenetic trees. The setting is challenging due to the discrete geometric structure of the trees themselves, which results in a nonlinear, non-smooth tree space, as well as size and dimensionality aspects (datasets can be very large, trees themselves can also be large). We propose a tree space based on a variant of algebraic geometry and study its structural and geometric characteristics in comparison to the current standard; in particular, we discuss their implications on statistics and data analysis. We will present results of a comparative study on influenza data.