Increasing trees have been extensively studied, since it is a simple model for many natural phenomena. Our paper focuses on sub-families of increasing trees. We measure the number of connected components obtained after having removed the nodes whose labels are smaller than a given value. This measure of cut-length allows, for example, to analyse in average an algorithm for tree-labelling. It is noticeable that we give exact formulae for the distribution of trees according to their size and cut-lengths. Our approach is based on a construction using grafting processes.