This is an announcement for the paper "Trees, linear orders and G\^ateaux smooth norms" by Richard J. Smith. Abstract: 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. Archive classification: math.FA Mathematics Subject Classification: 46B03; 46B26 Remarks: A different version of this paper is to appear in J. London Math. Soc The source file(s), arxiv12-10-07.tex: 60917 bytes, is(are) stored in gzipped form as 0710.4230.gz with size 18kb. The corresponding postcript file has gzipped size 102kb. Submitted from: rjs209@cam.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0710.4230 or http://arXiv.org/abs/0710.4230 or by email in unzipped form by transmitting an empty message with subject line uget 0710.4230 or in gzipped form by using subject line get 0710.4230 to: math@arXiv.org.