This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many mportant tree families that have applications in computer science or are used as probabilistic models in various applications, like \emphrecursive trees, heap-ordered trees or \emphbinary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters \textitdepth of a randomly chosen node, \textitdistance between two randomly chosen nodes, and the generalisations where \textitp nodes are randomly chosen Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures.