The Isoperimetric Problem in Regular Trees

We investigate the inner vertex-isoperimetric problem on the $d$-regular tree $T_d$. We first determine the exact value of the inner vertex-isoperimetric profile $I_d(k) = \min\{ |\partial D| \mid D\subset T_d \text{ finite and connected},\ |D|=k \}$, and we then introduce a boundary invariant, called the boundary branching excess $\tau(D)$, and show that it provides a simple criterion for optimality. A domain $D\subset T_d$ is shown to be isoperimetrically optimal if and only if $\tau(D)\le d-2$. Finally, we show that every domain in $T_d$ admits a canonical decomposition as an iterated gluing of full domains, namely domains whose entire boundary consists of leaves. This yields a complete description of all inner vertex-isoperimetric minimizers in $T_d$.