One way to efficiently disseminate information in a P2P overlay is to rely on a spanning tree. However, in a tree, interior nodes support a much higher load than leaf nodes. Also, the failure of a single node can break the tree, impairing the reliability of the dissemination protocol. These problems can be addressed by using multiple trees, such that each node is interior in just a few trees and a leaf node in the remaining, the multiple trees approach allows to achieve load distribution and also to send redundant information for fault-tolerance. This paper proposes Thicket, a decentralized algorithm to efficiently build and maintain such multiple trees over a single unstructured overlay network. The algorithm has been implemented and is extensively evaluated using simulation in a P2P overlay with 10.000 nodes.