How does the number of trees with even order that contain a perfect matching behave asymptotically?

Question

I recently found a nice result for trees of even order that do not contain a perfect matching. This led me to wonder ‘how many’ trees have perfect matchings, asymptotically speaking. Is anything known about the ratio (number of trees of order 2n with a perfect matching)/(number of trees of order 2n) as n tends to infinity? Do almost all trees not contain a perfect matching? Probably any such result would be for labelled trees.

Answer 1

Each edge of a tree determines a pair of rooted trees, and we call these rooted trees limbs of the tree. Allen Schwenk proved many years ago that the number of trees on $n$ vertices that contain a given rooted tree as a limb depends only on the number of vertices in the limb, and using this he shows that the proportion of trees on $n$ vertices that do not contain a given limb goes to zero as $n$ increases.

So if we take our limb $L$ to be the path on three vertices rooted at its centre, then it follows that almost all tree on $n$ vertices contain $L$ as a limb, and so almost all trees do not have a perfect matching.

Answer 2

The sequence $a(n)$ of the number of trees with $2n$ vertices that do have a perfect matching is this one. From the linked OEIS page you can figure that $$a(n) \sim c d^n/ n^{5/2}$$ where

$$c \approx 0.1128580768964135711615258 \quad \mbox{and} \quad d \approx 5.646542616232949712892713$$