Let \([2n] = \{1, \dots, 2n\}\). Then there are \((2n)^{2n-2}\) labelled trees on \([2n]\). If \([2n]\) is divided into two sets of equal cardinality, say \(L\) and \(D\), then exactly \(n^{2n-2}\) trees have the property that each edge joins a vertex in \(L\) to a vertex in \(D\). Independently choose \(r\) trees from among these trees. \(P_ r (n)\) is defined as the probability that the graph on \([2n]\), that is obtained by taking the union of the edges of the \(r\) trees, contains a perfect matching. It is well known that \(P_ 2(n) \to 1\) as \(n \to \infty\). The main result of the paper is \(1-P_ 2 (n)=O ((\log n)/n)\). Additional, a lower bound is given in \(1-P_ 2(n) > c/n\) with a desired constant \(c>0\) for all sufficiently large \(n\).