As broadcasting is widely used for miscellaneous maintenance operations in wireless ad hoc networks, where energy is a scarce resource, an efficient broadcasting protocol is of prime importance. One of the best known algorithm, named BIP (Broadcast Incremental Power), constructs a spanning tree rooted at a given node. This protocol offers very good results in terms of energy savings, but its computation is unfortunately centralized, as the source node needs to know the entire topology of the network to compute the tree. Many localized protocols have since been proposed, but none of them has ever reached the performances of BIP. Even distributed versions of the latter have been proposed, but they require a huge transmission overhead for information exchange and thus waste energy savings obtained thanks to the efficiency of the tree. In this paper, we propose and analyze a localized version of this protocol. In our method, each node is aware of the position of all the hosts in the set of its 2-hop neighborhood and compute the BIP tree on this set, based on information provided by the node from which it got the packet. That is, a tree is incrementally built thanks to information passed from node to node in the broadcast packet. Only the source node computes an initially empty tree to initiate the process. We also provide experimental results showing that this new protocol has performances very close to other good ones for low densities, and is very energy-efficient for higher densities with performances that equal the ones of BIP.