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a(k) is Heuberger and Wagner's G_k at lemma 6.2 (2). They show (theorem 3.3 (1)) that the largest number of maximum matchings in a tree of 7k+1 vertices is a(k+1) and there is a unique free tree with this many maximum matchings. (See A333347 for all tree sizes.) - Kevin Ryde, Apr 11 2020

This constant is Heuberger and Wagner's lambda. They consider the number of maximum matchings a tree of n vertices may have, and show that the largest number of maximum matchings (A333347) grows as O(lambda^(n/7)) (see A333346 for the 7th root). Lambda is the larger eigenvalue of matrix M = [8,3/5,3] which is raised to a power when counting matchings in a chain of "C" parts in the trees (their lemma 6.2).

Apart from the first digit the same as A176522. - R. J. Mathar, Apr 03 2020

First differences of A190872.

Heuberger and Wagner consider the number of maximum matchings a tree of n vertices may have. They show that the largest number of maximum matchings (A333347) grows as O(1.3916...^n) where the power is the constant here. This arises in their tree forms since each 7-vertex "C" part increases the number of matchings by a factor of matrix M=[8,3/5,3] (lemma 6.2). The larger eigenvalue of M is their lambda = A333345 and so a factor of lambda for each 7 vertices.

Heuberger and Wagner consider how many different maximum matchings a tree of n vertices may have. They determine the unique tree (free tree) of n vertices with the largest number of maximum matchings, or at n=6 and n=34 the two trees with equal largest number. a(n) is the matching number of the unique tree, and of both n=34 trees since they have the same matching number. For n=6, a(6)=1 is the star-6 which is their T_{6,1}. The other n=6 is their T_{6,2} and its matching number would be a(6)=2 instead.

The trees n!=2 have all pairs of leaves an even distance apart (the type of free tree counted by A304867). Vertices an even distance to a leaf are Heuberger and Wagner's type A, and vertices an odd distance to a leaf are type B. Per their definitions (and for any "even distance leaves" tree in fact), all type B vertices must be matched in a maximum matching and consequently the matching number is the number of type B vertices. 2n/7 appears in the formula below since each "C" part contains 7 vertices of which 2 are type B; then there are certain fixed additional B vertices according to n mod 7.