In 1963, Kesten proved a Pattern Theorem for self-avoiding walks, which says that any finite sequence of steps that can occur in the middle of a long self-avoiding walk must in fact occur pretty often on almost all self-avoiding walks. This result has had many applications in the theory of self-avoiding walks, including several ratio limit theorems. This talk will describe an analogous theorem for lattice trees and lattice animals. A weighted version of this theorem also applies to polymer collapse and percolation models. Applications and open questions will be discussed.