In tropical algebraic geometry, the zero sets of polynomial equations

are piecewise-linear. The tropical variety of a polynomial ideal is

the "logarithmic limit set" which was introduced by Bergman in 1971

and studied further by Bieri and Groves in the 1980's. We show that it

is embedded as a polyhedral subcomplex in the Gröbner fan of the

ideal. Our main example is the Grassmannian of lines in tropical

projective space. We interpret tropical lines as phylogenetic trees,

and we identify the Grassmannian with the space of trees of Billera,

Holmes and Vogtmann. The references are math.AG/0306366 and

math.AG/0304218 .

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