Bijections for Cayley trees, spanning trees, and their q-analogues

Abstract

We construct a family of extremely simple bijections that yield Cayley's famous formula for counting trees. The weight preserving properties of these bijections furnish a number of multivariate generating functions for weighted Cayley trees. Essentially the same idea is used to derive bijective proofs and q-analogues for the number of spanning trees of other graphs, including the complete bipartite and complete tripartite graphs. These bijections also allow the calculation of explicit formulas for the expected number of various statistics on Cayley trees.

Local extrema in random trees

“…such pairs ((i 1 , j 1 ,k 1 ),(1, j 2 ,k 2 )) total (for 4 elements in {2, ...,n}, there are 2 + 1 = 3 pairs with j 1 > j 2 = i 1 or j 1 > k 1 > i 1 = j 2 for 8 3 ; and there are 2 + 2 + 1 = 5 pairs with largest elements j 1 , j 2 or j 1 > k 1 > j 2 for 8 4 ). Observe that T contains S 8i ( (1, j1,k1),(i2, j2,k2)) if and only if T contains S 8i+2 ((i2, j2,k2),(1, j1,k1)) for i = 1,2.…”

“…Generating functions for the number of labelled trees of several types according to the number of ascents and descents are given in [4]. A functional equation satisfied by the generating function for the number of labelled trees according to the number of descents and leaves is given in [5].…”

A recursive algorithm for trees and forests

“…Eǧecioǧlu and Remmel [9] found a bijective proof of the enumeration of bipartite trees and tripartite trees and later they [10] found a bijection for the k-partite trees. The reader is encouraged to find a simple proof of Austin's formula by modifying our recursive algorithm for trees and forests.…”

Parallel Algorithms for Encoding and Decoding Blob Code

“…Namely, the ϑ n bijection by Egecioglu and Remmel [12]; the code due to Kreweras-Moszkowski [19]; the Chen code [9]; the Blob code, the Happy code, and the Dandelion code due to Picciotto [26]; and the MHappy code due to Caminiti and Petreschi [6]. In the last years, almost all these codes have been reinterpreted in a unified framework where their behavior is described as a transformation of the tree into a functional digraph [6].…”