Let $T$ be the set of vertices of a tree. We assume
that the Green function is finite and $G(s,t) \rightarrow 0$ as $|s|
\rightarrow \infty$ for each vertex $t$. For $v$ positive superharmonic
on $T$ and $E$ a subset of $T$, the reduced function of $v$ on $E$ is
the pointwise infimum of the set of positive superharmonic functions
that majorize $v$ on $E$. We give an explicit formula for the reduced
function in case $E$ is finite as well as several applications of this
formula. We define the minimal fine filter corresponding to each
boundary point of the tree and prove a tree version of the
Fatou-Naïm-Doob limit theorem, which involves the existence of
limits at boundary points following the minimal fine filter of the
quotient of a positive superharmonic by a positive harmonic function. We
deduce from this a radial limit theorem for such functions. We prove a
growth result for positive superharmonic functions from which we deduce
that, if the trees has transition probabilities all of which lie between
$\delta$ and $1/2-\delta$ for some $\delta \in (0,1/2)$ (for example
homogeneous trees with isotropic transition probabilities), then any
real-valued function on T which has a limit at a boundary point
following the minimal fine filter necessarily has a nontangential limit
there. We give an example of a tree for which minimal fine limits do
not imply nontangential limits, even for positive superharmonic
functions. Motivated by work on potential theory on halfspaces and
Brelot spaces, we define the harmonic fine filter corresponding to each
boundary point of the tree. In contrast to the classical setting, we are
able to show that it is the same as the minimal fine filter.