We consider the group $$\mathcal G$$ of isometries of a semi-homogeneous tree $$T=T_{q_+,q_-}$$ with valencies $$q_+ +1$$ and $$q_- +1$$ and its two orbits, respectively $$V_+$$ and $$V_-$$ , on the set of vertices. We make use of the action of $$\mathcal G$$ to equip each of $$V_\pm $$ with a convolution product, hence with a notion of positive definite functions. The $$\ell ^1$$ -functions radial around a root vertex $$v_{ 0}$$ in, say, $$V_+$$ form an abelian convolution algebra. We study its multiplicative functionals, called spherical functions, that are 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 $$\mathcal G$$ ; in this way, we produce a series of unitary spherical representations. For $$q_+