  We introduce two notions of convexity for an infinite regular tree. For these
two notions we show that given a continuous boundary datum there exists a
unique convex envelope on the tree and characterize the equation that this
envelope satisfies. We also relate the equation with two versions of the
Laplacian on the tree. Moreover, for a function defined on the tree, the convex
envelope turns out to be the solution to the obstacle problem for this
equation.
