Prüfer multiplicities for a full rooted binary tree

A tree on $2m+1$ labels has one root of degree two, $m-1$ other interior vertices of degree three, and $m+1$ leaves precisely when its Prüfer code contains one label once, $m-1$ labels twice, and all other labels zero times. The Prüfer degree formula gives both directions, and identifies the root as the uniquely once-occurring label. Children are unordered unless a plane embedding is additional data.