A357701 Irregular triangle read by rows where row n is the vertex depths of the rooted binary tree with Colijn-Plazzotta tree number n, traversed in pre-order, numerically larger child first.

0, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 2, 1, 2, 2, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 3, 2, 1, 2, 2, 0, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2, 0, 1, 2, 3, 3, 2, 3, 3, 1, 0, 1, 2, 3, 3, 2, 3, 3, 1, 2, 2, 0, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 0, 1, 2, 3, 3, 2, 3, 3, 1, 2, 3, 3, 2, 3, 3

Offset: 1

Author

Kevin Ryde, Oct 11 2022

Keywords

• nonn
• easy
• tabf

Comments

Colijn and Plazzotta enumerate rooted binary trees (every vertex 0 or 2 children) by n=1 as a singleton or if n>1 then a root with child subtrees x = A002024(n-1) and y = A002260(n-1), which is y = 1..x for each successive x.

Depth levels are distance down from the root, so 0 for the root itself, 1 for children of the root, and so on.

The pre-order traversal visits a vertex and then recursively traverses its "x" subtree followed by its "y" subtree.

The resulting depths vector is the lexicographically greatest among all possible orderings of siblings (as seen by induction).

Rows are in lexicographically increasing order (again by induction) so that an equivalent definition is greatest depths vectors in increasing order.

Row n has length A064002(n) which is the number of vertices.

Row n begins 0,1,2,...,h where h is the height of the tree, i.e. greatest depth of any vertex.

Examples

			Triangle begins:
      k=1  2  3  4  5  6  7  8  9 10 11
  n=1:  0,
  n=2:  0, 1, 1,
  n=3:  0, 1, 2, 2, 1,
  n=4:  0, 1, 2, 2, 1, 2, 2,
  n=5:  0, 1, 2, 3, 3, 2, 1,
  n=6:  0, 1, 2, 3, 3, 2, 1, 2, 2,
  n=7:  0, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2,
  n=8:  0, 1, 2, 3, 3, 2, 3, 3, 1,
  n=9:  0, 1, 2, 3, 3, 2, 3, 3, 1, 2, 2,
For n=6, tree 6 is as follows, with vertices numbered by pre-order traversal (column number k),
          1         depth=0
        /   \
      2       7     depth=1
     / \     / \
    3   6   8   9   depth=2
   / \
  4  5              depth=3
  row(6) = depths 0,1,2,3,3,2,1,2,2
		

Links

• Kevin Ryde, Table of n, a(n) for rows 1..500, flattened
• Caroline Colijn and Giacomo Plazzotta, A Metric on Phylogenetic Tree Shapes, Systematic Biology, volume 67, number 1, January 2018, pages 113-126.
• Kevin Ryde, PARI/GP Code

Crossrefs

Cf. A064002 (row lengths), A357702 (row sums).

Cf. A002024 (larger child), A002260 (smaller child).

Programs

Mathematica

yList=FoldList[{#1,#2}&,1,Range[2,20]]//Flatten;x[n_]:=Floor[Sqrt[2*n]+1/2];y[n_]:=yList[[n]];row[1]={0};row[n_]:=row[n]={0}~Join~(row[x[n-1]]+1)~Join~(row[y[n-1]]+1);Flatten[Array[row,11]] (* Shenghui Yang, Apr 15 2025 *)

PARI

\\ See links.

Formula

row(n) = {0, row(x)+1, row(y)+1} for n>=2, where row(c)+1 means +1 on each term of row c, and where x = A002024(n-1) and y = A002260(n-1).