In this paper we present a substantially general theory of parenthesis-free notations for finite plane trees. We obtain stronger one-to-oneness results, including a characterization of one-to-oneness for a large class of notations, and a quite general sufficient condition for one-to-oneness that involves the recursive structure of plane trees in what appears to be a minimal way. We then study various properties of notations that promise to be of practical interest. The two most important of these—single scan bottom-up readability, and single scan top-down readability—both turn out to be particular cases of our general sufficient condition for one-to-oneness. We formulate simple descriptions of all notations that have one or the other of these properties, and find a simple transformation of notations that establishes a one-to-one correspondence between them. The unique fixed point of this transformation turns out to be the only notation single scan readable in both directions.