Between Brackets

Brackets are symbols showing the structure of an expression in a linear notation. A term can be represented as a finite ordered tree of which each vertex is labeled with a member of a given set of non-logical constant or function symbols. This labeled tree can be embedded in the plane in such a way that the vertices correspond to labeled points on a circle and the edges to non-crossing line segments. In this article, we show that in the standard notation (with each subterm between brackets) the matching left and right brackets can be interpreted as edges of the (circular) dual of this plane tree. We also present a dual notation in which the matching brackets correspond to the edges of the plane tree itself. Finally, we present a mixed notation in which constants are separated by exactly two brackets. We define these notations for terms in which arguments are split into left and right arguments. This corresponds to outerplanar trees in which child nodes are split into left and right children. In Polish notation, brackets are avoided by using each constant as a prefix operator with a fixed number of arguments. We generalize this to infix operators with fixed numbers of left and right arguments.