  We introduce a linearly ordered set Z and use it to prove a necessity
condition for the existence of a G\^ateaux smooth norm on C(T), where T is a
tree. This criterion is directly analogous to the corresponding equivalent
condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits
a G\^ateaux smooth lattice norm then it also admits a lattice norm with
strictly convex dual norm.
