Tree topologies,which construct spatial graphs with large characteristic path lengths and small clustering coefficients,are ubiquitous in deployments of wireless sensor networks.Small worlds are investigated in tree-based networks.Due to link additions,characteristic path lengths reduce rapidly and clustering coefficients increase greatly.A tree abstract,Cayley tree,is considered for the study of the navigation algorithm,which runs automatically in the small worlds of tree-based networks.In the further study,epidemics in the small worlds of tree-based wireless sensor networks on the large scale are studied,and the percolation threshold is calculated,at which the outbreak of the epidemic takes place.Compared with Cayley tree,there is a smaller percolation threshold suffering from the epidemic.