Irreducible components of Hilbert schemes of rational curves with given normal bundle

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a reducible Hilbert scheme parametrizing all the rational curves in P-s with a given decomposition type of the normal bundle. We also characterize smooth nondegenerate rational curves contained in rational normal scrolls in terms of the splitting type of their restricted tangent bundles and compute their normal bundles.