  On a semi-homogeneous tree, we study the $\ell^p$-spectrum of the Laplace
operator $\mu_1$ (the isotropic nearest-neighbor transition operator); the
known results in the much simpler setting of homogeneous trees are obtained as
particular cases. The spectrum is given by the eigenvalues of spherical
functions, i.e., eigenfunctions of $mu_1$ that are radial with respect to a
reference vertex $v_0$ and normalized there. We show that spherical functions
are boundary integrals of generalized Poisson kernels that, unlike the
homogeneous setting, are not complex powers of the usual Poisson kernel. We
compute these generalized Poisson kernels via Markov chains and their
generating functions, whence we work out explicit expressions for spherical
functions. On semi-homogeneous trees, spherical functions turn out to have an
$\ell^p$ behavior that does not occur on homogeneous trees: one of them, for an
appropriate choice of $v_0$, belongs to $\ell^p$ for some $p<2$.
  Up to normalization, the operator $\mu_1^2$ differs from the step-2 Laplacian
$\mu_2$ only by a shift. On the other hand, the recurrence relation associated
to the semi-homogeneous $\mu_2$ is that of a polygonal graph, akin to that of
the Laplacian in a homogeneous tree. By this token, we compute the spectra of
$\mu_1^2$ on a semi-homogeneous tree, hence, by extracting square roots, the
$\ell^p$-spectrum of $\mu_1$ for $1\leqslant p <\infty$, and show that it is
disconnected for $p$ in an interval containing 2 but it is connected at all
other values of $p$.
