In this paper we consider the heat semigroup {Wt}t>0 defined by the combinatorial Laplacian and two subordinated families of {Wt}t>0 on homogeneous trees X. We characterize the weights u on X for which the pointwise convergence to initial data of the above families holds for every f ∈ L p (X, µ, u) with 1 ≤ p < ∞, where µ represents the counting measure in X . We prove that this convergence property in X is equivalent to the fact that the maximal operator on t ∈ (0, R), for some R > 0, defined by the semigroup is bounded from L p (X, µ, u) into L p (X, µ, v) for some weight v on X.