  We consider the equiprobable distribution of spanning trees on the square
lattice. All bonds of each tree can be oriented uniquely with respect to an
arbitrary chosen site called the root. The problem of predecessors is finding
the probability that a path along the oriented bonds passes sequentially fixed
sites $i$ and $j$. The conformal field theory for the Potts model predicts the
fractal dimension of the path to be 5/4. Using this result, we show that the
probability in the predecessors problem for two sites separated by large
distance $r$ decreases as $P(r) \sim r^{-3/4}$. If sites $i$ and $j$ are
nearest neighbors on the square lattice, the probability $P(1)=5/16$ can be
found from the analytical theory developed for the sandpile model. The known
equivalence between the loop erased random walk (LERW) and the directed path on
the spanning tree says that $P(1)$ is the probability for the LERW started at
$i$ to reach the neighboring site $j$. By analogy with the self-avoiding walk,
$P(1)$ can be called the return probability. Extensive Monte-Carlo simulations
confirm the theoretical predictions.
