Binary trees with choosable edge lengths

For a set of two-element sets of non-negative real numbers we consider rooted strict binary trees with the property that the two edges leading from every non-leaf to its two children are assigned lengths l1 and l2 with {l1,l2}∈L.For choices of L for which l1+l2 is constant for every {l1,l2}∈L which models that a certain total length can be distributed with some degree of freedom specified by L to incident edges, we study the asymptotic growth of the maximum number of leaves of bounded depths in such trees and the existence of such trees with leaves at individually specified maximum depths.