In this paper we consider a graded Lie algebra, L, of finite depth m, and study the interplay between the depth of L and the growth of the integers dim Li. A subspace W in a graded vector space V is called full if for some integers d, N, q, dim Vk ≤ d ∑k + qi = k Wi, i ≥ N. We define an equivalence relation on the subspaces of V by U ∼ W if U and W are full in U + W. Two subspaces V, W in L are then called L-equivalent (V ∼L W) if for all ideals K ⊂ L, V ∩ K ∼ W ∩ K. Then our main result asserts that the set ℒ of L-equivalence classes of ideals in L is a distributive lattice with at most 2m elements. To establish this we show that for each ideal I there is a Lie subalgebra E ⊂ L such that E ∩ I = 0, E ⊕ I is full in L, and depth E + depth I ≤ depth L.