A356918 -id:A356918 - cp's OEIS frontend

A356917 Irregular triangle read by rows where row n lists the Colijn-Plazzotta subtree numbers, in ascending order, of each vertex of the rooted binary tree with their tree number n.

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 7, 1, 1, 1, 1, 1, 2, 2, 4, 8, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 9, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 10, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 11

Offset: 1

Kevin Ryde, Sep 19 2022

Colijn and Plazzotta enumerate rooted binary trees (every vertex 0 or 2 children) by n=1 as a singleton and thereafter tree n is a root with child subtrees x = A002024(n-1) and y = A002260(n-1).
Each row starts with 1's for the childless vertices (A064064(n) of them).
Each row ends with n itself (the tree root).
The second last term in each row is the numerically largest subtree of the root, which is x.
Row lengths are A064002(n), the number of vertices.

			Triangle begins:
      k=1  2  3  4  5  6  7  8  9 10 11
  n=1:  1,
  n=2:  1, 1, 2,
  n=3:  1, 1, 1, 2, 3,
  n=4:  1, 1, 1, 1, 2, 2, 4,
  n=5:  1, 1, 1, 1, 2, 3, 5,
  n=6:  1, 1, 1, 1, 1, 2, 2, 3, 6,
  n=7:  1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 7,
  n=8:  1, 1, 1, 1, 1, 2, 2, 4, 8,
Tree n=6 and its subtree numbers are as follows and row 6 is these subtree numbers in ascending order.
          6  root
        /   \
      3       2
     / \     / \
    2   1   1   1
   / \
  1   1

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, see section 2.3 where their L_n = row n here.
Kevin Ryde, PARI/GP Code

Cf. A002024, A002260 (root subtrees).
Cf. A064002 (number of vertices), A064064 (number of childless).
Cf. A356918 (d1 metric).

PARI
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row(n) = sort {row(x), row(y), n} for n>=2, where x = A002024(n-1) and y = A002260(n-1).

cp's OEIS Frontend

A356917 Irregular triangle read by rows where row n lists the Colijn-Plazzotta subtree numbers, in ascending order, of each vertex of the rooted binary tree with their tree number n.

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