1. The mathematician's tree is a collection of pairs of elements, conveniently called points. A pair of points is a link, and a collection of links is a tree if any two of the points from which the links are formed can be connected by a chain, and if there are no closed chains. A tree is determined if there is provided a complete schedule of the links composing the tree. Conversely, no description of a tree can be adequate unless a complete schedule is recoverable from the description. An arbitrary schedule of links does not usually describe a tree, and the object of this note is to give canonical schedules for the most general finite tree, and to specify these schedules as economically as possible. The fundamental result established is that the structure of a tree with n points can be encoded in a row of n − 2 symbols each of which may stand for any one of the points. Cayley's theorem, that the number of different trees with n given points is nn−2, is an immediate corollary.