  We consider the group $\mathrm{Aut}(T)$ of isometries of a semi-homogeneous
tree $T=T_{q_+,q_-}$ with valencies $q_+ +1$ and $q_- +1$ and its two orbits
$V_+$, $V_-$ respectively. We make use of the action of $\mathrm{Aut} (T)$ to
equip the spaces of finitely supported radial functions on each of $V_\pm$ with
convolution products, hence with a notion of positive definite functions. The
$\ell^1$-functions radial around a root vertex $v_0\in V_+$ form an abelian
convolution algebra. We study its multiplicative functionals, called spherical
functions, given by eigenfunctions of the nearest-neighbor isotropic transition
operator (the Laplace operator on $T$, and determine which of them are positive
definite. Each positive definite function gives rise to a unitary
representation of $\mathrm{Aut}(T)$; in this way, we produce a series of
unitary spherical representations. For $q_+<q_-$, the representation whose
spherical function has eigenvalue 0 is square-integrable.
