Tree codes that preserve increases and degree sequences

Abstract We define a bijection from the set n of rooted Cayley's trees with n vertices to the set [1,n]n−1 of words of n − 1 letters written with the alphabet [1,n]. This code has three remarkable properties: (1) As in Prufer's code [1], the degree of every vertex is visible in the word mT= mT(1)…mT(n−1) coding the tree T. More precisely, if di denotes the degree of the vertex i: If i is the root of T, then i appears di times in mT. If not, then i appears di − 1 times in mT. (2) For any rooted tree, increasing (or decreasing) edges can be defined in the following way: let (i,j) be an edge of T such that the minimal path joining i to the root passes through j; then the edge (i,j) is said to be increasing if j>i (and decreasing if j i. We prove that the set of letters i in mT such that mT(i)>i corresponds to the set of increasing edges of T. c(i)>i⇔mT(i)>i. (3) From the code mT one derives a code m(T) for Cayley's trees which ‘preserves’ degrees and increases of edges. The code m(T) can be defined independently of mT and therefore from it one derives a bijective proof of Cayley's formula [6]. These properties yield new generating functions. The bijections generalize the classic bijection between permutations linking cycles to outstanding elements [4] (while the bijections of Egecioglu and Remmel [2], which have similar properties, may be viewed as variants of Joyal's code [5]).