  We investigate projective varieties which are geometric models of binary
symmetric phylogenetic 3-valent trees. We prove that these varieties have
Gorenstein terminal singularities (with small resolution) and they are Fano
varieties of index 4. Moreover any two such varieties associated to trees with
the same number of leaves are deformation equivalent, that is, they are in the
same connected component of the Hilbert scheme of the projective space. As an
application we provide a simple formula for computing their Hilbert-Ehrhard
polynomial.
