Almost every tree with n edges decomposes

Let T be a tree with m edges. We show that if T has at least ( m + 1 ) / 3 leaves then T decomposes the complete bipartite graph K 2 n , 2 n . In particular this shows that almost all trees decompose K 2 n , 2 n , a result connected to a conjecture of Graham and Hنggkvist.