Λ-trees and their applications

To most mathematicians and computer scientists the word "tree" conjures up, in addition to the usual image, the image of a connected graph with no circuits. We shall deal with various aspects and generalizations of these mathematical trees. (As Peter Shalen has pointed out, there will be leaves and foliations in this discussion, but they do not belong to the trees!) In the last few years various types of trees have been the subject of much investigation. But this activity has not been exposed much to the wider mathematical community. To me the subject is very appealing for it mixes very nave geometric considerations with the very sophisticated geometric and algebraic structures. In fact, part of the drama of the subject is guessing what type of techniques will be appropriate for a given investigation: Will it be direct and simple notions related to schematic drawings of trees or will it be notions from the deepest parts of algebraic group theory, ergodic theory, or commutative algebra which must be brought to bear? Part of the beauty of the subject is that the naive tree considerations have an impact on these more sophisticated topics. In addition, trees form a bridge between these disparate subjects.