  A tree $T$ is said to be homogeneous if it is uniquely rooted and there
exists an integer $b\meg 2$, called the branching number of $T$, such that
every $t\in T$ has exactly $b$ immediate successors. A vector homogeneous tree
$\mathbf{T}$ is a finite sequence $(T_1,...,T_d)$ of homogeneous trees and its
level product $\otimes\mathbf{T}$ is the subset of the cartesian product
$T_1\times ...\times T_d$ consisting of all finite sequences $(t_1,...,t_d)$ of
nodes having common length.
  We study the behavior of measurable events in probability spaces indexed by
the level product $\otimes\mathbf{T}$ of a vector homogeneous tree
$\mathbf{T}$. We show that, by refining the index set to the level product
$\otimes\mathbf{S}$ of a vector strong subtree $\bfcs$ of $\mathbf{S}$, such
families of events become highly correlated. An analogue of Lebesgue's density
Theorem is also established which can be considered as the "probabilistic"
version of the density Halpern--L\"{a}uchli Theorem.
