A family of trees with no uncountable branches

We construct a family of 2(N1) trees of size N-1 and no uncountable branches that in a certain way codes all omega(1)-sequences of infinite subsets of omega. This coding allows us to conclude that in the presence of the club guessing between N-1 and No, these trees are pairwise very different. In such circumstances we can also conclude that the universality number of the ordered class of trees of size N-1 with no uncountable branches under "metric-preserving" reductions must be at least the continuum. From the topological point of view, the above results show that under the same assumptions there are 2(N1) pairwise non-isometrically embeddable first countable omega(1)-metric spaces with a strong non-ccc property, and that their universality number under isometric embeddings is at least the continuum. Without the non-ccc requirement, a family of 2(N1) pairwise non-isometrically embeddable first countable omega(1)-metric spaces exists in ZFC by an earlier result of S. Todorcevic. The set-theoretic assumptions mentioned above are satisfied in many natural models of set theory (such as the ones obtained after forcing by a ccc forcing over a model of lozenge). We use a similar method to discuss trees of size kappa with no uncountable branches, for any regular uncountable kappa.