A bijective enumeration of labeled trees with given indegree sequence

Abstract

For a labeled tree on the vertex set { 1 , 2 , … , n } , the local direction of each edge ( i j ) is from i to j if i < j . For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ = 1 e 1 2 e 2 … of a tree on the vertex set { 1 , 2 , … , n } is a partition of n − 1 . We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.