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In this paper we consider the heat semigroup $\{W_t\}_{t>0}$ defined by thecombinatorial Laplacian and two subordinated families of $\{W_t\}_{t>0}$ onhomogeneous trees $X$. We characterize the weights $u$ on $X$ for which thepointwise convergence to initial data of the above families holds for every$f\in L^{p}(X,\mu,u)$ with $1\le p<\infty$, where $\mu$ represents the countingmeasure in $X$ . We prove that this convergence property in $X$ is equivalentto the fact that the maximal operator on $t\in (0,R)$, for some $R>0$, definedby the semigroup is bounded from $L^{p}(X,\mu,u)$ into $L^{p}(X,\mu,v)$ forsome weight $v$ on $X$.

Pointwise convergence of the heat and subordinates of the heat  semigroups associated with the Laplace operator on homogeneous trees and two  weighted $L^p$ maximal inequalities