  We systematically develop the multivariable counterpart of the theory of
weighted shifts on rooted directed trees. Capitalizing on the theory of product
of directed graphs, we introduce and study the notion of multishifts on
directed Cartesian product of rooted directed trees. This framework unifies the
theory of weighted shifts on rooted directed trees and that of classical
unilateral multishifts. Moreover, this setup brings into picture some new
phenomena such as the appearance of system of linear equations in the
eigenvalue problem for the adjoint of a multishift. In the first half of the
paper, we focus our attention mostly on the multivariable spectral theory and
function theory including finer analysis of various joint spectra and wandering
subspace property for multishifts. In the second half, we separate out two
special classes of multishifts, which we refer to as torally balanced and
spherically balanced multishifts. The classification of these two classes is
closely related to toral and spherical polar decompositions of multishifts.
Furthermore, we exhibit a family of spherically balanced multishifts on
$d$-fold directed Cartesian product $\mathscr T$ of rooted directed trees.
These multishifts turn out be multiplication $d$-tuples $\mathscr M_{z, a}$ on
certain reproducing kernel Hilbert spaces $\mathscr H_a$ of vector-valued
holomorphic functions defined on the unit ball $\mathbb B^d$ in $\mathbb C^d$,
which can be thought of as tree analogs of the multiplication $d$-tuples acting
on the reproducing kernel Hilbert spaces associated with the kernels
$\frac{1}{(1-\langle{z},{{w}\rangle})^a}~(z, w \in \mathbb B^d, a \in \mathbb
N).$
