Random increasing k-trees represent an interesting, useful class of strongly dependent graphs that have been studied widely, including being used recently as models for complex networks. We study in this paper an informative notion called connectivity-profile and derive, by several analytic means, asymptotic estimates for its expected value, together with the limiting distribution in certain cases; some interesting consequences predicting more precisely the shapes of random k-trees are also given. Our methods of proof rely essentially on a bijection between k-trees and ordinary trees, and the resolution of a linear system.