We consider broadcasting a message from one node of a tree to all other nodes. In the presence of up to k link failures the tree becomes disconnected, and only nodes in the connected component C containing the source can be informed. The maximum ratio between the time used by a broadcasting scheme B to inform C and the optimal time to inform C, taken over all components C yielded by configurations of at most k faults, is the k-vulnerability of B. This is the maximum slowdown incurred by B due to the lack of a priori knowledge of fault location, for at most k faults. This measure of fault tolerance is similar to the competitive factor of on-line algorithms: in both cases, the performance of an algorithm lacking some crucial information is compared to the performance of an off-line'' algorithm, one that is given this information as input. It is also the first known tool to measure and compare fault tolerance of broadcasting schemes in trees. We seek broadcasting schemes with low vulnerability, working for tree networks. It turns out that schemes that give the best broadcasting time in a fault-free environment may have very high vulnerability, i.e., poor fault tolerance, for some trees. The main result of this paper is an algorithm that, given an arbitrary tree T and an integer k, computes a broadcasting scheme B with lowest possible k-vulnerability among all schemes working for T. Our algorithm has running time O(kn 2 +n 2 log n), where n is the size of the tree. We also give an algorithm to find a universally fault-tolerant'' broadcasting scheme in a tree T: one that approximates the lowest possible k-vulnerability, for all k simultaneously.