Let $\mathbb{T}_d$ be a homogeneous tree in which every vertex has
$d$ neighbors. A new proof is given that the contact process on
$\mathbb{T}_d$ exhibits two phase transitions when $d \geq 3$, a behavior which
distinguishes it from the contact process on $\mathbb{Z}^n$. This is the first
proof which does not involve calculation of bounds on critical values, and it
is much shorter than the previous proof for the binary tree, $\mathbb{T}_3$.
The method is extended to prove the existence of an intermediate phase for a
more general class of trees with exponential growth and certain symmetry
properties, for which no such result was previously known.