论文信息 - How to encode a tree

How to encode a tree

We construct bijections giving three "codes" for trees. These codes follow naturally from the Matrix Tree Theorem of Tutte and have many advantages over the one produced by Prufer in 1918. One algorithm gives explicitly a bijection that is implicit in Orlin's manipulatorial proof of Cayley's formula (the formula was actually found first by Borchardt). Another is based on a proof of Knuth. The third is an implementation of Joyal's pseudo-bijective proof of the formula, and is equivalent to one previously found by Egecioglu and Remmel. In each case, we have at least two algorithms, one of which involves hands-on manipulations of the tree while the other involves a combinatorial and linear algebraic manipulation of a matrix.

S. Picciotto

[1] C. W. Borchardt. Ueber eine der Interpolation entsprechende Darstellung der Eliminations-Resultante. , 1860 .

[2] W. T. Tutte. The dissection of equilateral triangles into equilateral triangles , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[3] Donald E. Knuth,et al. Oriented subtrees of an arc digraph , 1967 .

[4] James B. Orlin,et al. Line-digraphs, arborescences, and theorems of tutte and knuth , 1978, J. Comb. Theory, Ser. B.

[5] A. Joyal. Une théorie combinatoire des séries formelles , 1981 .

[6] S. Milne,et al. Method for constructing bijections for classical partition identities. , 1981, Proceedings of the National Academy of Sciences of the United States of America.

[7] S. Chaiken. A Combinatorial Proof of the All Minors Matrix Tree Theorem , 1982 .

[8] Doron Zeilberger,et al. A combinatorial approach to matrix algebra , 1985, Discret. Math..

[9] Ömer Egecioglu,et al. Bijections for Cayley trees, spanning trees, and their q-analogues , 1986, J. Comb. Theory, Ser. A.

[10] D. White,et al. Constructive combinatorics , 1986 .

[11] A. Cayley. A theorem on trees , 2009 .