If $X$ and $Y$ are two reasonable spaces with universal covers $\tilde{X}$ and $\tilde{Y}$, there is a nice picture of the universal cover $\widetilde{X \wedge vee Y}$ which has the combinatorial pattern of an infinite tree. The tree is bipartite with vertices labeled by the symbols $X$ and $Y$. The edges from an $X$ vertex are bijective with the fundamental group $\pi_1(X)$, and likewise for $Y$ vertices and $\pi_1(Y)$. To make $\widetilde{X \wedge vee Y}$, replace each $X$ vertex by $\tilde{X}$ and each $Y$ vertex by $\tilde{Y}$. The base point of $X$ lifts to $\pi_1(X)$ |\pi_1(X)|$points in$\tilde{X}$, and likewise for$Y$. In$\widetilde{X \wedge vee Y}$, copies of$\tilde{X}$are attached to copies of$\tilde{Y}$at lifts of base points. For example, if$X = Y = \mathbb{R}P^2$, then the tree is an infinite chain and$\widetilde{X \wedge vee Y}$is an infinite chain of 2-spheres. This tree picture nicely and dramatically generalizes to Bass-Serre theory. 1 If$X$and$Y$are two reasonable spaces with universal covers$\tilde{X}$and$\tilde{Y}$, there is a nice picture of the universal cover$\widetilde{X \wedge Y}$which has the combinatorial pattern of an infinite tree. The tree is bipartite with vertices labeled by the symbols$X$and$Y$. The edges from an$X$vertex are bijective with the fundamental group$\pi_1(X)$, and likewise for$Y$vertices and$\pi_1(Y)$. To make$\widetilde{X \wedge Y}$, replace each$X$vertex by$\tilde{X}$and each$Y$vertex by$\tilde{Y}$. The base point of$X$lifts to$\pi_1(X)$points in$\tilde{X}$, and likewise for$Y$. In$\widetilde{X \wedge Y}$, copies of$\tilde{X}$are attached to copies of$\tilde{Y}$at lifts of base points. For example, if$X = Y = \mathbb{R}P^2$, then the tree is an infinite chain and$\widetilde{X \wedge Y}\$ is an infinite chain of 2-spheres.