Dense subtrees in complete Boolean algebras

Every good introduction to set-theoretic forcing starts with basic forcing algebras like ”adding a Cohen real” or ”adding a Cohen subset of ω1”. It seems that these algebras are particularly simple mainly because they contain a dense subtree, so we might as well pretend we are forcing with a tree. Such a forcing is especially easy to visualize as the generic object will be a branch through the tree and the notions of comparability and compatibility coincide. The purpose of this note is to give a useful characterization of forcing algebras that contain such a dense subtree. As far as our knowledge goes, the following characterization has not been pointed out anywhere in the literature even though it seriously helps determining the isomorphism type of a forcing algebra.