We consider a continuous space which models the set of all phylogenetic trees having a fixed set of leaves. This space has a natural metric of nonpositive curvature, giving a way of measuring distance between phylogenetic trees and providing some procedures for averaging or combining several trees whose leaves are identical. This geometry also shows which trees appear within a fixed distance of a given tree and enables construction of convex hulls of a set of trees. This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.  2001 Elsevier Science

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We introduce the property of k-decomposability for simplicial complexes which, if satisfied by the dual simplicial complex of a convex polyhedron, implies that the diameter of the polyhedron is bounded by a polynomial of degree k + 1 in the number of facets. These properties form a hierarchy, each one implying the next. The strongest, vertex decomposability, implies the Hirsch conjecture; the weakest is equivalent to shellability. We show that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest. We conclude with a strengthened form of the property of shellability which would imply the Hirsch conjecture for polytopes.

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A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant• defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, • is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid, • is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight, • behaves simply under matroid duality, • has a simple expansion in terms of P -partition enumerators, and • is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising in work of Lafforgue, where lack of such a decomposition implies the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis.

We define an equivalence relation on integer compositions and show that two ribbon Schur functions are identical if and only if their defining compositions are equivalent in this sense. This equivalence is completely determined by means of a factorization for compositions: equivalent compositions have factorizations that differ only by reversing some of the terms. As an application, we can derive identities on certain Littlewood-Richardson coefficients.Finally, we consider the cone of symmetric functions having a nonnnegative representation in terms of the fundamental quasisymmetric basis. We show the Schur functions are among the extremes of this cone and conjecture its facets are in bijection with the equivalence classes of compositions.