Geometry of weighted recursive and affine preferential attachment trees

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n?2 new un is added to its parent chosen randomly according some rule. In weighted tree, we choose uk among {u1,u2,…,un?1} with probability proportional wk, where (wn)n?1 deterministic sequence that fix beforehand. affine preferential attachment fitnesses, choosing any ak+deg+(uk), deg+(uk) denotes current number children, fitnesses (an)n?1 as parameter model. show for (an)n?1, corresponding has same distribution random weights (with explicit distribution). then prove almost sure scaling limit convergences statistics associated trees goes infinity, such degree sequence, height, profile also weak convergence measures carried on tree. Thanks connection between these results apply