Universal covers of wedge sums

Let \(X = X_1\vee\cdots\vee X_n\), where each \(X_i\) is path-connected, locally path-connected and semilocally simply connected, and each basepoint is a deformation retract of an open neighborhood. The universal cover \(\tilde X\) is a union of copies of the universal covers \(\tilde X_i\), glued at lifts of the basepoint, such that the graph with one vertex for each copy, one vertex for each lift of the basepoint, and an edge whenever a copy contains a lift, is a tree. For example, the universal cover of \(\RP^2\vee\RP^2\) is an infinite chain of \(2\)-spheres, each tangent to its two neighbors [@Hat02].