Interdependence between carathйodory numbers and $n$-distributivity in lattices

For a lattice $L$ with zero a subset $F\subseteq L$ is called a (lower) spanning tree if for any y $y\in L/\{0\}$ there exists $x\in F$ such that $0<x\leqslant y$. The main goal of the present note is a proof of two theorems, one of which is the following:
THEOREM 1. Suppose that the spanning tree of an algebraic lattice $L$ consists of completely join-irreducible elements and that each element $x\in L$ is the union of some subset (in general, infinite) of $F$. Then the Caratheodory number of $L$ relative to the spanning tree $F$ is equal to the distributivity number of this lattice.
The second theorem states the same result as the first, though under different conditions on the lattice $L$ and the spanning tree $F$.