  Let $\mathcal{T}$ be a locally finite tree whose geometric boundary has
infinitely many points. Suppose that a non-amenable group $\G$ acts
isometrically and geometrically on the tree $\mathcal{T}$.
  In this paper, we show that if the length spectrum is Diophantine, then there
exists a continuous function $C$ on $\mathcal{T}^2$ such that the heat kernel
$p(t,x,y)$ of $\mathcal{T}$ satisfies
  $$\lim_{t\rightarrow \infty}t^{3/2}e^{\lambda_0t}p(t,x,y)=C(x,y)$$
  for any $x,y\in \mathcal{T}$. Here, $\lambda_0$ is the bottom of the spectrum
of the Laplacian on $\mathcal{T}$.
