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This article presents a method for finding the critical probability $p_c$ forthe Bernoulli bond percolation on graphs with the so-called tree-likestructure. Such a graph can be decomposed into a tree of pieces, each of whichhas finitely many isomorphism classes. This class of graphs includes the Cayleygraphs of amalgamated products, HNN extensions or general groups acting ontrees. It also includes all transitive graphs with more than one end. The ideaof the method is to find a multi-type Galton--Watson branching process (with aparameter $p$) which has finite expected population size if and only if theexpected percolation cluster size is finite. This provides sufficientinformation about $p_c$. In particular, if the pairwise intersections of piecesare finite, then $p_c$ is the smallest positive $p$ such that $\operatorname{det}(M-1)=0$, where $M$ is the first-moment matrix of the branching process.If the pieces of the tree-like structure are finite, then $p_c$ is an algebraicnumber and we give an algorithm computing $p_c$ as a root of some algebraicfunction. We show that any Cayley graph of a virtually free group (i.e., agroup acting on a tree with finite vertex stabilizers) with respect to anyfinite generating set has a tree-like structure with finite pieces. Inparticular, we show how to compute $p_c$ for the Cayley graph of a free groupwith respect to any finite generating set.

Critical percolation of virtually free groups and other tree-like graphs