This paper presents an on-line algorithm for the construction of position trees, i.e. an algorithm which constructs the position tree for a given string while reading the string from left to right. In addition, an on-line correction algorithm is presented which - upon a change in the string - can be used to construct the new position tree. Moreover, special attention is paid to computers with small memory. Compactification of the trees and transport costs between main and secondary storage are discussed.