The resiliency of a network is its ability to remain \emph{effectively} functioning also when any of its nodes or links fails. However, to reduce operational and set-up costs, a network should be small in size, and this conflicts with the requirement of being resilient. In this paper we address this trade-off for the prominent case of the {\em broadcasting} routing scheme, and we build efficient (i.e., sparse and fast) \emph{fault-tolerant approximate shortest-path trees}, for both the edge and vertex \emph{single-failure} case. In particular, for an $n$-vertex non-negatively weighted graph, and for any constant $\varepsilon >0$, we design two structures of size $O(\frac{n \log n}{\varepsilon^2})$ which guarantee $(1+\varepsilon)$-stretched paths from the selected source also in the presence of an edge/vertex failure. This favorably compares with the currently best known solutions, which are for the edge-failure case of size $O(n)$ and stretch factor 3, and for the vertex-failure case of size $O(n \log n)$ and stretch factor 3. Moreover, we also focus on the unweighted case, and we prove that an ordinary $(α,β)$-spanner can be slightly augmented in order to build efficient fault-tolerant approximate \emph{breadth-first-search trees}.