Counting embeddings of a chain into a tree

For a given finite poset T whose Hasse diagram is a tree and its maximal element is its root, we compare the number of those embeddings into T of a chain of a given length which contain the maximal element of T and the number of those embeddings which do not contain the maximal element of T. For a given positive integer k, we establish average depth thresholds for T beyond which there are always more embeddings of the second kind than those of the first one. We, actually, give a better than 1 upper bound for the ratio between the numbers of these embeddings that depends on the structure of T.