  We study the watermelon probabilities in the uniform spanning forests on the
two-dimensional semi-infinite square lattice near either open or closed
boundary to which the forests can or cannot be rooted, respectively. We derive
universal power laws describing the asymptotic decay of these probabilities
with the distance between the reference points growing to infinity, as well as
their non-universal constant prefactors. The obtained exponents match with the
previous predictions made for the related dense polymer models using the
Coulomb Gas technique and Conformal Field Theory, as well as with the lattice
calculations made by other authors in different settings. We also discuss the
logarithmic corrections some authors argued to appear in the watermelon
correlation functions on the infinite lattice. We show that the full account
for diverging terms of the lattice Green function, which ensures the correct
probability normalization, provides the pure power law decay in the case of
semi-infinite lattice with closed boundary studied here, as well as in the case
of infinite lattice discussed elsewhere. The solution is based on the
all-minors generalization of the Kirchhoff matrix tree theorem, the image
method and the developed asymptotic expansion of the Kirchhoff determinants.
